combination and permutation
机械专业英语词汇(整理篇)
机械专业英语词汇陶瓷ceramics合成纤维synthetic fibre电化学腐蚀electrochemical corrosion 车架automotive chassis悬架suspension转向器redirector变速器speed changer板料冲压sheet metal parts孔加工spot facing machining车间workshop工程技术人员engineer气动夹紧pneuma lock数学模型mathematical model画法几何descriptive geometry机械制图Mechanical drawing投影projection视图view剖视图pro标准件standard component零件图part drawing装配图assembly drawing尺寸标注size marking技术要求technical requirements刚度rigidity内力internal force位移displacement截面section疲劳极限fatigue limit断裂fracture塑性变形plastic distortion脆性材料brittleness material刚度准则rigidity criterion垫圈washer垫片spacer直齿圆柱齿轮straight toothed spur gear斜齿圆柱齿轮helical-spur gear直齿锥齿轮straight bevel gear运动简图kinematic sketch齿轮齿条pinion and rack蜗杆蜗轮worm and worm gear虚约束passive constraint曲柄crank摇杆racker凸轮cams共轭曲线conjugate curve范成法generation method定义域definitional domain值域range导数\\微分differential coefficient求导derivation定积分definite integral不定积分indefinite integral曲率curvature偏微分partial differential毛坯rough游标卡尺slide caliper千分尺micrometer calipers攻丝tap二阶行列式second order determinant 逆矩阵inverse matrix线性方程组linear equations概率probability随机变量random variable排列组合permutation and combination气体状态方程equation of state of gas 动能kinetic energy势能potential energy机械能守恒conservation of mechanical energy动量momentum桁架truss轴线axes余子式cofactor逻辑电路logic circuit触发器flip-flop脉冲波形pulse shape数模digital analogy液压传动机构fluid drive mechanism 机械零件mechanical parts淬火冷却quench淬火hardening回火tempering调质hardening and tempering磨粒abrasive grain结合剂bonding agent砂轮grinding wheel后角clearance angle龙门刨削planing主轴spindle主轴箱headstock卡盘chuck加工中心machining center车刀lathe tool车床lathe钻削镗削bore车削turning磨床grinder基准benchmark钳工locksmith锻forge压模stamping焊weld拉床broaching machine拉孔broaching装配assembling铸造found流体动力学fluid dynamics流体力学fluid mechanics加工machining液压hydraulic pressure切线tangent机电一体化mechanotronics mechanical-electrical integration 气压air pressure pneumatic pressure 稳定性stability介质medium液压驱动泵fluid clutch液压泵hydraulic pump阀门valve失效invalidation强度intensity载荷load应力stress安全系数safty factor可靠性reliability螺纹thread螺旋helix键spline销pin滚动轴承rolling bearing滑动轴承sliding bearing弹簧spring制动器arrester brake十字结联轴节crosshead联轴器coupling链chain皮带strap精加工finish machining粗加工rough machining变速箱体gearbox casing腐蚀rust氧化oxidation磨损wear耐用度durability随机信号random signal离散信号discrete signal超声传感器ultrasonic sensor集成电路integrate circuit挡板orifice plate残余应力residual stress套筒sleeve扭力torsion冷加工cold machining电动机electromotor汽缸cylinder过盈配合interference fit热加工hotwork摄像头CCD camera倒角rounding chamfer优化设计optimal design工业造型设计industrial moulding design有限元finite element滚齿hobbing插齿gear shaping伺服电机actuating motor铣床milling machine钻床drill machine镗床boring machine步进电机stepper motor丝杠screw rod导轨lead rail组件subassembly可编程序逻辑控制器Programmable Logic Controller PLC电火花加工electric spark machining 电火花线切割加工electrical discharge wire - cutting相图phase diagram热处理heat treatment固态相变solid state phase changes有色金属nonferrous metal陶瓷ceramics合成纤维synthetic fibre电化学腐蚀electrochemical corrosion 车架automotive chassis悬架suspension转向器redirector变速器speed changer板料冲压sheet metal parts孔加工spot facing machining车间workshop工程技术人员engineer气动夹紧pneuma lock数学模型mathematical model画法几何descriptive geometry机械制图Mechanical drawing投影projection视图view剖视图pro标准件standard component零件图part drawing装配图assembly drawing尺寸标注size marking技术要求technical requirements刚度rigidity内力internal force位移displacement截面section疲劳极限fatigue limit断裂fracture塑性变形plastic distortion脆性材料brittleness material刚度准则rigidity criterion垫圈washer垫片spacer直齿圆柱齿轮straight toothed spur gear斜齿圆柱齿轮helical-spur gear直齿锥齿轮straight bevel gear运动简图kinematic sketch齿轮齿条pinion and rack蜗杆蜗轮worm and worm gear虚约束passive constraint曲柄crank摇杆racker凸轮cams共轭曲线conjugate curve范成法generation method定义域definitional domain值域range导数\\微分differential coefficient求导derivation定积分definite integral不定积分indefinite integral曲率curvature偏微分partial differential毛坯rough游标卡尺slide caliper千分尺micrometer calipers攻丝tap二阶行列式second order determinant 逆矩阵inverse matrix线性方程组linear equations概率probability随机变量random variable排列组合permutation and combination气体状态方程equation of state of gas 动能kinetic energy势能potential energy机械能守恒conservation of mechanical energy动量momentum桁架truss轴线axes余子式cofactor逻辑电路logic circuit触发器flip-flop脉冲波形pulse shape数模digital analogy液压传动机构fluid drive mechanism 机械零件mechanical parts淬火冷却quench淬火hardening回火tempering调质hardening and tempering磨粒abrasive grain结合剂bonding agent砂轮grinding wheelAssembly line 组装线Layout 布置图Conveyer 流水线物料板Rivet table 拉钉机Rivet gun 拉钉枪Screw driver 起子Pneumatic screw driver 气动起子worktable 工作桌OOBA 开箱检查fit together 组装在一起fasten 锁紧(螺丝)fixture 夹具(治具)pallet 栈板barcode 条码barcode scanner 条码扫描器fuse together 熔合fuse machine热熔机repair修理operator作业员QC品管supervisor 课长ME 制造工程师MT 制造生技cosmetic inspect 外观检查inner parts inspect 内部检查thumb screw 大头螺丝lbs. inch 镑、英寸EMI gasket 导电条front plate 前板rear plate 后板chassis 基座bezel panel 面板power button 电源按键reset button 重置键Hi-pot test of SPS 高源高压测试Voltage switch of SPS 电源电压接拉键sheet metal parts 冲件plastic parts 塑胶件SOP 制造作业程序material check list 物料检查表work cell 工作间trolley 台车carton 纸箱sub-line 支线left fork 叉车personnel resource department 人力资源部production department生产部门planning department企划部QC Section品管科stamping factory冲压厂painting factory烤漆厂molding factory成型厂common equipment常用设备uncoiler and straightener整平机punching machine 冲床robot机械手hydraulic machine油压机lathe车床planer |plein|刨床miller铣床grinder磨床linear cutting线切割electrical sparkle电火花welder电焊机staker=reviting machine铆合机position职务president董事长general manager总经理special assistant manager特助factory director厂长department director部长deputy manager | =vice manager副理section supervisor课长deputy section supervisor =vice section superisor副课长group leader/supervisor组长line supervisor线长assistant manager助理to move, to carry, to handle搬运be put in storage入库pack packing包装to apply oil擦油to 锉毛刺final inspection终检to connect material接料to reverse material 翻料wet station沾湿台Tiana天那水cleaning cloth抹布to load material上料to unload material卸料to return material/stock to退料scraped |\\'skr?pid|报废scrape ..v.刮;削deficient purchase来料不良manufacture procedure制程deficient manufacturing procedure制程不良oxidation |\\' ksi\\'dei?n|氧化scratch刮伤dents压痕defective upsiding down抽芽不良defective to staking铆合不良embedded lump镶块feeding is not in place送料不到位stamping-missing漏冲production capacity生产力education and training教育与训练proposal improvement提案改善spare parts=buffer备件forklift叉车trailer=long vehicle拖板车compound die合模die locker锁模器pressure plate=plate pinch压板bolt螺栓administration/general affairs dept总务部automatic screwdriver电动启子thickness gauge厚薄规gauge(or jig)治具power wire电源线buzzle蜂鸣器defective product label不良标签identifying sheet list标示单location地点present members出席人员subject主题conclusion结论decision items决议事项responsible department负责单位pre-fixed finishing date预定完成日approved by / checked by / prepared by核准/审核/承办PCE assembly production schedule sheet PCE组装厂生产排配表model机锺work order工令revision版次remark备注production control confirmation生产确认checked by初审approved by核准department部门stock age analysis sheet 库存货龄分析表on-hand inventory现有库存available material良品可使用obsolete material良品已呆滞to be inspected or reworked 待验或重工total合计cause description原因说明part number/ P/N 料号type形态item/group/class类别quality品质prepared by制表notes说明year-end physical inventory difference analysis sheet 年终盘点差异分析表physical inventory盘点数量physical count quantity帐面数量difference quantity差异量cause analysis原因分析raw materials原料materials物料finished product成品semi-finished product半成品packing materials包材good product/accepted goods/ accepted parts/good parts良品defective product/non-good parts不良品disposed goods处理品warehouse/hub仓库on way location在途仓oversea location海外仓spare parts physical inventory list备品盘点清单spare molds location模具备品仓skid/pallet栈板tox machine自铆机wire EDM线割EDM放电机coil stock卷料sheet stock片料tolerance工差score=groove压线cam block滑块pilot导正筒trim剪外边pierce剪内边drag form压锻差pocket for the punch head挂钩槽slug hole废料孔feature die公母模expansion dwg展开图radius半径shim(wedge)楔子torch-flame cut火焰切割set screw止付螺丝form block折刀stop pin定位销round pierce punch=die button圆冲子shape punch=die insert异形子stock locater block定位块under cut=scrap chopper清角active plate活动板baffle plate挡块cover plate盖板male die公模female die母模groove punch压线冲子air-cushion eject-rod气垫顶杆spring-box eject-plate弹簧箱顶板bushing block衬套insert 入块club car高尔夫球车capability能力parameter参数factor系数phosphate皮膜化成viscosity涂料粘度alkalidipping脱脂main manifold主集流脉bezel斜视规blanking穿落模dejecting顶固模demagnetization去磁;消磁high-speed transmission高速传递heat dissipation热传rack上料degrease脱脂rinse水洗alkaline etch龄咬desmut剥黑膜D.I. rinse纯水次Chromate铬酸处理Anodize阳性处理seal封孔revision版次part number/P/N料号good products良品scraped products报放心品defective products不良品finished products成品disposed products处理品barcode条码flow chart流程表单assembly组装stamping冲压molding成型spare parts=buffer备品coordinate座标dismantle the die折模auxiliary fuction辅助功能poly-line多义线heater band 加热片thermocouple热电偶sand blasting喷沙grit 砂砾derusting machine除锈机degate打浇口dryer烘干机induction感应induction light感应光response=reaction=interaction感应ram连杆edge finder巡边器concave凸convex凹short射料不足nick缺口speck瑕??shine亮班splay 银纹gas mark焦痕delamination起鳞cold slug冷块blush 导色gouge沟槽;凿槽satin texture段面咬花witness line证示线patent专利grit沙砾granule=peuet=grain细粒grit maker抽粒机cushion缓冲magnalium镁铝合金magnesium镁金metal plate钣金lathe车mill锉plane刨grind磨drill铝boring镗blinster气泡fillet镶;嵌边through-hole form通孔形式voller pin formality滚针形式cam driver铡楔shank摸柄crank shaft曲柄轴augular offset角度偏差velocity速度production tempo生产进度现状torque扭矩spline=the multiple keys花键quenching淬火tempering回火annealing退火carbonization碳化tungsten high speed steel钨高速的moly high speed steel钼高速的organic solvent有机溶剂bracket小磁导liaison联络单volatile挥发性resistance电阻ion离子titrator滴定仪beacon警示灯coolant冷却液crusher破碎机阿基米德蜗杆Archimedes worm安全系数safety factor; factor of safety安全载荷safe load凹面、凹度concavity扳手wrench板簧flat leaf spring半圆键woodruff key变形deformation摆杆oscillating bar摆动从动件oscillating follower摆动从动件凸轮机构cam with oscillating follower摆动导杆机构oscillating guide-bar mechanism摆线齿轮cycloidal gear摆线齿形cycloidal tooth profile摆线运动规律cycloidal motion摆线针轮cycloidal-pin wheel包角angle of contact保持架cage背对背安装back-to-back arrangement背锥back cone ；normal cone背锥角back angle背锥距back cone distance比例尺scale比热容specific heat capacity闭式链closed kinematic chain闭链机构closed chain mechanism臂部arm变频器frequency converters变频调速frequency control of motor speed变速speed change变速齿轮change gear change wheel变位齿轮modified gear变位系数modification coefficient标准齿轮standard gear标准直齿轮standard spur gear表面质量系数superficial mass factor表面传热系数surface coefficient of heat transfer表面粗糙度surface roughness并联式组合combination in parallel并联机构parallel mechanism并联组合机构parallel combined mechanism并行工程concurrent engineering并行设计concurred design, CD不平衡相位phase angle of unbalance不平衡imbalance (or unbalance)不平衡量amount of unbalance不完全齿轮机构intermittent gearing波发生器wave generator波数number of waves补偿compensation参数化设计parameterization design, PD 残余应力residual stress操纵及控制装置operation control device 槽轮Geneva wheel 槽轮机构Geneva mechanism ；Maltese cross槽数Geneva numerate槽凸轮groove cam侧隙backlash差动轮系differential gear train差动螺旋机构differential screw mechanism差速器differential常用机构conventional mechanism; mechanism in common use车床lathe承载量系数bearing capacity factor承载能力bearing capacity成对安装paired mounting尺寸系列dimension series齿槽tooth space齿槽宽spacewidth齿侧间隙backlash齿顶高addendum齿顶圆addendum circle齿根高dedendum齿根圆dedendum circle齿厚tooth thickness齿距circular pitch齿宽face width齿廓tooth profile齿廓曲线tooth curve齿轮gear齿轮变速箱speed-changing gear boxes齿轮齿条机构pinion and rack齿轮插刀pinion cutter; pinion-shaped shaper cutter齿轮滚刀hob ,hobbing cutter齿轮机构gear齿轮轮坯blank齿轮传动系pinion unit齿轮联轴器gear coupling齿条传动rack gear齿数tooth number齿数比gear ratio齿条rack齿条插刀rack cutter; rack-shaped shaper cutter齿形链、无声链silent chain齿形系数form factor齿式棘轮机构tooth ratchet mechanism插齿机gear shaper重合点coincident points重合度contact ratio冲床punch传动比transmission ratio, speed ratio传动装置gearing; transmission gear传动系统driven system传动角transmission angle传动轴transmission shaft串联式组合combination in series串联式组合机构series combined mechanism串级调速cascade speed control创新innovation creation创新设计creation design垂直载荷、法向载荷normal load唇形橡胶密封lip rubber seal磁流体轴承magnetic fluid bearing从动带轮driven pulley从动件driven link, follower从动件平底宽度width of flat-face从动件停歇follower dwell从动件运动规律follower motion从动轮driven gear粗线bold line粗牙螺纹coarse thread大齿轮gear wheel打包机packer打滑slipping带传动belt driving带轮belt pulley带式制动器band brake单列轴承single row bearing单向推力轴承single-direction thrust bearing单万向联轴节single universal joint单位矢量unit vector当量齿轮equivalent spur gear; virtual gear当量齿数equivalent teeth number; virtual number of teeth 当量摩擦系数equivalent coefficient of friction当量载荷equivalent load刀具cutter导数derivative倒角chamfer导热性conduction of heat导程lead导程角lead angle等加等减速运动规律parabolic motion; constant acceleration and deceleration motion等速运动规律uniform motion; constant velocity motion等径凸轮conjugate yoke radial cam等宽凸轮constant-breadth cam等效构件equivalent link等效力equivalent force等效力矩equivalent moment of force等效量equivalent等效质量equivalent mass等效转动惯量equivalent moment of inertia等效动力学模型dynamically equivalent model底座chassis低副lower pair点划线chain dotted line（疲劳）点蚀pitting垫圈gasket垫片密封gasket seal碟形弹簧belleville spring顶隙bottom clearance定轴轮系ordinary gear train; gear train with fixed axes动力学dynamics动密封kinematical seal动能dynamic energy动力粘度dynamic viscosity动力润滑dynamic lubrication动平衡dynamic balance动平衡机dynamic balancing machine动态特性dynamic characteristics动态分析设计dynamic analysis design动压力dynamic reaction动载荷dynamic load端面transverse plane端面参数transverse parameters端面齿距transverse circular pitch端面齿廓transverse tooth profile端面重合度transverse contact ratio端面模数transverse module端面压力角transverse pressure angle锻造forge对称循环应力symmetry circulating stress对心滚子从动件radial (or in-line ) roller follower对心直动从动件radial (or in-line ) translating follower对心移动从动件radial reciprocating follower对心曲柄滑块机构in-line slider-crank (or crank-slider) mechanism多列轴承multi-row bearing多楔带poly V-belt多项式运动规律polynomial motion多质量转子rotor with several masses惰轮idle gear额定寿命rating life额定载荷load ratingII 级杆组dyad发生线generating line发生面generating plane法面normal plane法面参数normal parameters法面齿距normal circular pitch法面模数normal module法面压力角normal pressure angle法向齿距normal pitch法向齿廓normal tooth profile法向直廓蜗杆straight sided normal worm 法向力normal force反馈式组合feedback combining反向运动学inverse ( or backward) kinematics反转法kinematic inversion反正切Arctan 范成法generating cutting仿形法form cutting方案设计、概念设计concept design, CD防振装置shockproof device飞轮flywheel飞轮矩moment of flywheel非标准齿轮nonstandard gear非接触式密封non-contact seal非周期性速度波动aperiodic speed fluctuation非圆齿轮non-circular gear粉末合金powder metallurgy分度线reference line; standard pitch line分度圆reference circle; standard (cutting) pitch circle分度圆柱导程角lead angle at reference cylinder分度圆柱螺旋角helix angle at reference cylinder分母denominator分子numerator分度圆锥reference cone; standard pitch cone分析法analytical method封闭差动轮系planetary differential复合铰链compound hinge复合式组合compound combining复合轮系compound (or combined) gear train复合平带compound flat belt复合应力combined stress复式螺旋机构Compound screw mechanism复杂机构complex mechanism杆组Assur group干涉interference刚度系数stiffness coefficient刚轮rigid circular spline钢丝软轴wire soft shaft刚体导引机构body guidance mechanism 刚性冲击rigid impulse (shock)刚性转子rigid rotor刚性轴承rigid bearing刚性联轴器rigid coupling高度系列height series高速带high speed belt高副higher pair格拉晓夫定理Grashoff`s law根切undercutting公称直径nominal diameter高度系列height series功work工况系数application factor工艺设计technological design工作循环图working cycle diagram工作机构operation mechanism工作载荷external loads工作空间working space工作应力working stress工作阻力effective resistance工作阻力矩effective resistance moment 公法线common normal line公共约束general constraint公制齿轮metric gears功率power功能分析设计function analyses design共轭齿廓conjugate profiles共轭凸轮conjugate cam构件link鼓风机blower固定构件fixed link; frame固体润滑剂solid lubricant关节型操作器jointed manipulator惯性力inertia force惯性力矩moment of inertia ,shaking moment惯性力平衡balance of shaking force惯性力完全平衡full balance of shaking force惯性力部分平衡partial balance of shaking force惯性主矩resultant moment of inertia惯性主失resultant vector of inertia冠轮crown gear广义机构generation mechanism广义坐标generalized coordinate轨迹生成path generation 轨迹发生器path generator滚刀hob滚道raceway滚动体rolling element滚动轴承rolling bearing滚动轴承代号rolling bearing identification code滚针needle roller滚针轴承needle roller bearing滚子roller滚子轴承roller bearing滚子半径radius of roller滚子从动件roller follower滚子链roller chain滚子链联轴器double roller chain coupling滚珠丝杆ball screw滚柱式单向超越离合器roller clutch过度切割undercutting函数发生器function generator函数生成function generation含油轴承oil bearing耗油量oil consumption耗油量系数oil consumption factor赫兹公式H. Hertz equation合成弯矩resultant bending moment合力resultant force合力矩resultant moment of force黑箱black box横坐标abscissa互换性齿轮interchangeable gears花键spline滑键、导键feather key滑动轴承sliding bearing滑动率sliding ratio滑块slider环面蜗杆toroid helicoids worm环形弹簧annular spring缓冲装置shocks; shock-absorber灰铸铁grey cast iron回程return回转体平衡balance of rotors混合轮系compound gear train积分integrate机电一体化系统设计mechanical-electrical integration system design机构mechanism机构分析analysis of mechanism机构平衡balance of mechanism机构学mechanism机构运动设计kinematic design of mechanism机构运动简图kinematic sketch of mechanism机构综合synthesis of mechanism机构组成constitution of mechanism机架frame, fixed link机架变换kinematic inversion机器machine机器人robot机器人操作器manipulator机器人学robotics技术过程technique process技术经济评价technical and economic evaluation技术系统technique system机械machinery机械创新设计mechanical creation design, MCD机械系统设计mechanical system design, MSD机械动力分析dynamic analysis of machinery机械动力设计dynamic design of machinery机械动力学dynamics of machinery机械的现代设计modern machine design机械系统mechanical system机械利益mechanical advantage机械平衡balance of machinery机械手manipulator机械设计machine design; mechanical design机械特性mechanical behavior机械调速mechanical speed governors机械效率mechanical efficiency机械原理theory of machines and mechanisms 机械运转不均匀系数coefficient of speed fluctuation机械无级变速mechanical stepless speed changes基础机构fundamental mechanism基本额定寿命basic rating life基于实例设计case-based design,CBD基圆base circle基圆半径radius of base circle基圆齿距base pitch基圆压力角pressure angle of base circle 基圆柱base cylinder基圆锥base cone急回机构quick-return mechanism急回特性quick-return characteristics急回系数advance-to return-time ratio急回运动quick-return motion棘轮ratchet棘轮机构ratchet mechanism棘爪pawl极限位置extreme (or limiting) position 极位夹角crank angle between extreme (or limiting) positions计算机辅助设计computer aided design, CAD计算机辅助制造computer aided manufacturing, CAM计算机集成制造系统computer integrated manufacturing system, CIMS计算力矩factored moment; calculation moment计算弯矩calculated bending moment加权系数weighting efficient加速度acceleration加速度分析acceleration analysis加速度曲线acceleration diagram尖点pointing; cusp尖底从动件knife-edge follower间隙backlash间歇运动机构intermittent motion mechanism减速比reduction ratio减速齿轮、减速装置reduction gear减速器speed reducer减摩性anti-friction quality渐开螺旋面involute helicoid渐开线involute渐开线齿廓involute profile渐开线齿轮involute gear渐开线发生线generating line of involute 渐开线方程involute equation渐开线函数involute function渐开线蜗杆involute worm渐开线压力角pressure angle of involute 渐开线花键involute spline简谐运动simple harmonic motion键key键槽keyway交变应力repeated stress交变载荷repeated fluctuating load交叉带传动cross-belt drive交错轴斜齿轮crossed helical gears胶合scoring角加速度angular acceleration角速度angular velocity角速比angular velocity ratio角接触球轴承angular contact ball bearing角接触推力轴承angular contact thrust bearing角接触向心轴承angular contact radial bearing角接触轴承angular contact bearing铰链、枢纽hinge校正平面correcting plane接触应力contact stress接触式密封contact seal阶梯轴multi-diameter shaft结构structure结构设计structural design截面section节点pitch point节距circular pitch; pitch of teeth节线pitch line节圆pitch circle节圆齿厚thickness on pitch circle节圆直径pitch diameter节圆锥pitch cone 节圆锥角pitch cone angle解析设计analytical design紧边tight-side紧固件fastener径节diametral pitch径向radial direction径向当量动载荷dynamic equivalent radial load径向当量静载荷static equivalent radial load径向基本额定动载荷basic dynamic radial load rating径向基本额定静载荷basic static radial load tating径向接触轴承radial contact bearing径向平面radial plane径向游隙radial internal clearance径向载荷radial load径向载荷系数radial load factor径向间隙clearance静力static force静平衡static balance静载荷static load静密封static seal局部自由度passive degree of freedom矩阵matrix矩形螺纹square threaded form锯齿形螺纹buttress thread form矩形牙嵌式离合器square-jaw positive-contact clutch绝对尺寸系数absolute dimensional factor绝对运动absolute motion绝对速度absolute velocity均衡装置load balancing mechanism抗压强度compression strength开口传动open-belt drive开式链open kinematic chain开链机构open chain mechanism可靠度degree of reliability可靠性reliability可靠性设计reliability design, RD空气弹簧air spring空间机构spatial mechanism空间连杆机构spatial linkage空间凸轮机构spatial cam空间运动副spatial kinematic pair空间运动链spatial kinematic chain空转idle宽度系列width series框图block diagram雷诺方程Reynolds‘s equation离心力centrifugal force离心应力centrifugal stress离合器clutch离心密封centrifugal seal理论廓线pitch curve理论啮合线theoretical line of action隶属度membership力force力多边形force polygon力封闭型凸轮机构force-drive (or force-closed) cam mechanism力矩moment力平衡equilibrium力偶couple力偶矩moment of couple连杆connecting rod, coupler连杆机构linkage连杆曲线coupler-curve连心线line of centers链chain链传动装置chain gearing链轮sprocket sprocket-wheel sprocket gear chain wheel联组V 带tight-up V belt联轴器coupling shaft coupling两维凸轮two-dimensional cam临界转速critical speed六杆机构six-bar linkage龙门刨床double Haas planer轮坯blank轮系gear train螺杆screw螺距thread pitch螺母screw nut螺旋锥齿轮helical bevel gear螺钉screws 螺栓bolts螺纹导程lead螺纹效率screw efficiency螺旋传动power screw螺旋密封spiral seal螺纹thread (of a screw)螺旋副helical pair螺旋机构screw mechanism螺旋角helix angle螺旋线helix ,helical line绿色设计green design design for environment马耳他机构Geneva wheel Geneva gear 马耳他十字Maltese cross脉动无级变速pulsating stepless speed changes脉动循环应力fluctuating circulating stress脉动载荷fluctuating load铆钉rivet迷宫密封labyrinth seal密封seal密封带seal belt密封胶seal gum密封元件potted component密封装置sealing arrangement面对面安装face-to-face arrangement面向产品生命周期设计design for product`s life cycle, DPLC名义应力、公称应力nominal stress模块化设计modular design, MD模块式传动系统modular system模幅箱morphology box模糊集fuzzy set模糊评价fuzzy evaluation模数module摩擦friction摩擦角friction angle摩擦力friction force摩擦学设计tribology design, TD摩擦阻力frictional resistance摩擦力矩friction moment摩擦系数coefficient of friction摩擦圆friction circle磨损abrasion wear; scratching末端执行器end-effector目标函数objective function耐腐蚀性corrosion resistance耐磨性wear resistance挠性机构mechanism with flexible elements挠性转子flexible rotor内齿轮internal gear内齿圈ring gear内力internal force内圈inner ring能量energy能量指示图viscosity逆时针counterclockwise (or anticlockwise)啮出engaging-out啮合engagement, mesh, gearing啮合点contact points啮合角working pressure angle啮合线line of action啮合线长度length of line of action啮入engaging-in牛头刨床shaper凝固点freezing point; solidifying point 扭转应力torsion stress扭矩moment of torque扭簧helical torsion spring诺模图NomogramO 形密封圈密封O ring seal盘形凸轮disk cam盘形转子disk-like rotor抛物线运动parabolic motion疲劳极限fatigue limit疲劳强度fatigue strength偏置式offset偏( 心) 距offset distance偏心率eccentricity ratio偏心质量eccentric mass偏距圆offset circle偏心盘eccentric偏置滚子从动件offset roller follower偏置尖底从动件offset knife-edge follower 偏置曲柄滑块机构offset slider-crank mechanism拼接matching评价与决策evaluation and decision频率frequency平带flat belt平带传动flat belt driving平底从动件flat-face follower平底宽度face width平分线bisector平均应力average stress平均中径mean screw diameter平均速度average velocity平衡balance平衡机balancing machine平衡品质balancing quality平衡平面correcting plane平衡质量balancing mass平衡重counterweight平衡转速balancing speed平面副planar pair, flat pair平面机构planar mechanism平面运动副planar kinematic pair平面连杆机构planar linkage平面凸轮planar cam平面凸轮机构planar cam mechanism平面轴斜齿轮parallel helical gears普通平键parallel key其他常用机构other mechanism in common use起动阶段starting period启动力矩starting torque气动机构pneumatic mechanism奇异位置singular position起始啮合点initial contact , beginning of contact气体轴承gas bearing千斤顶jack嵌入键sunk key强迫振动forced vibration切齿深度depth of cut曲柄crank曲柄存在条件Grashoff`s law曲柄导杆机构crank shaper (guide-bar)mechanism曲柄滑块机构slider-crank (or crank-slider) mechanism曲柄摇杆机构crank-rocker mechanism曲齿锥齿轮spiral bevel gear曲率curvature曲率半径radius of curvature曲面从动件curved-shoe follower曲线拼接curve matching曲线运动curvilinear motion曲轴crank shaft驱动力driving force驱动力矩driving moment (torque)全齿高whole depth权重集weight sets球ball球面滚子convex roller球轴承ball bearing球面副spheric pair球面渐开线spherical involute球面运动spherical motion球销副sphere-pin pair球坐标操作器polar coordinate manipulator燃点spontaneous ignition热平衡heat balance; thermal equilibrium人字齿轮herringbone gear冗余自由度redundant degree of freedom 柔轮flexspline柔性冲击flexible impulse; soft shock柔性制造系统flexible manufacturing system; FMS柔性自动化flexible automation润滑油膜lubricant film润滑装置lubrication device润滑lubrication润滑剂lubricant三角形花键serration spline三角形螺纹V thread screw三维凸轮three-dimensional cam三心定理Kennedy`s theorem砂轮越程槽grinding wheel groove砂漏hour-glass 少齿差行星传动planetary drive with small teeth difference设计方法学design methodology设计变量design variable设计约束design constraints深沟球轴承deep groove ball bearing生产阻力productive resistance升程rise升距lift实际廓线cam profile十字滑块联轴器double slider coupling; Oldham‘s coupling矢量vector输出功output work输出构件output link输出机构output mechanism输出力矩output torque输出轴output shaft输入构件input link数学模型mathematic model实际啮合线actual line of action双滑块机构double-slider mechanism, ellipsograph双曲柄机构double crank mechanism双曲面齿轮hyperboloid gear双头螺柱studs双万向联轴节constant-velocity (or double) universal joint双摇杆机构double rocker mechanism双转块机构Oldham coupling双列轴承double row bearing双向推力轴承double-direction thrust bearing松边slack-side顺时针clockwise瞬心instantaneous center死点dead point四杆机构four-bar linkage速度velocity速度不均匀( 波动) 系数coefficient of speed fluctuation速度波动speed fluctuation速度曲线velocity diagram速度瞬心instantaneous center of。
permutation formula
permutation formulapermutation formula is taking m (m n) elements from n different elements (different elements) in a certain order, which is called a arrangement of m elements from n different elements. The arrangement is related to the order of the elements, and the combination is independent of the order. The principle of addition and multiplication are the basis of arrangement and combination.step:Step 1, the first bit can be filled from one of the n elements, a total of n kinds of filling;In Step 2, the second place can only be filled in from any one of the remaining n-1 elements, with a total of n-1 filling method;In Step 3, the third digit can only fill in any one of the remaining n-2 elements, with a total of n-2 filling methods;……Step m, when the previous m-1 vacancies are filled, the m-1 position can only be filled from one of the remaining n- (m-1) elements, with a total of n-m + 1 filling method.According to the step-wise counting principle, all m vacancies are filled with n (n-1) (n-2)... (n-m + 1) filling.fundamental:The arrangement is related to the order of the elements, and the combination is independent of the order. For example, 231 and 213 are two permutations, the sum of 2 + 3 + 1 and the sum of 2 + 1 + 3 are a combination, and the two basic principles are the basis of the arrangement and combination.(1) Addition principle: do a thing, complete it can have n class method, in the first type method there are m1 different methods, in the second type method there are m2 different methods,..., in the n class method has mn different methods, then complete this thing a total of N=m1 + m2 + m3 +... + mn kinds of different methods.(2) Multiplication principle: do a thing, complete it needs to be divided into n steps, do the first step m1 different methods, do the second step m2 different methods,..., do n mn different methods, then complete this thing total N=m1 m2 m3... mn different methods.If there is an n class method, it is the classification problem, and the first class method isindependent. Therefore, we use the addition principle, and we must use the addition principle.。
IB必备 数学专业名词中英对照
数学mathematics公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.) 被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point 分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth powern次方the power of n, the nth power 开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth rootn次方根the root of n, the nth root 集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix 行列式determinant几何geometry点point线line 面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi(pl.), diamond正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon 圆circle圆心center半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter面积area轨迹locus, loca(pl.)相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis 圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incenter外心excenter旁心escenter垂心Orthocenter重心barycenter内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation比例proportion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution 图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagram。
高中数学排列组合公式
高中数学排列组合公式
排列组合（Permutation and Combination）是组合学最基本的概念。
所谓排列，就是指从给定个数的元素中取出指定个数的元素进行排序。
组合则是指从给定个数的元素中仅仅取出指定个数的元素，不考虑排序。
排列组合的中心问题是研究给定要求的排列和组合可能出现的情况总数。
扩展资料
排列的定义：从n个不同元素中，任取m(m≤n,m与n均为自然数,下同）个不同的元素按照一定的.顺序排成一列，叫做从n个不同元素中取出m个元素的一个排列；从n个不同元素中取出m(m≤n）个元素的所有排列的个数，叫做从n个不同元素中取出m个元素的排列数，用符号 A(n,m）表示。
计算公式：
此外规定0! = 1
组合的定义：从n个不同元素中，任取m(m≤n）个元素并成一组，叫做从n个不同元素中取出m个元素的一个组合；从n个不同元素中取出m(m≤n）个元素的所有组合的个数，叫做从n个不同元素中取出m个元素的组合数。
用符号 C(n,m) 表示。
计算公式：；C(n,m)=C(n,n-m）。
（n≥m)
其他排列与组合公式从n个元素中取出m个元素的循环排列数=A(n,m)/m=n!/m(n-m)!. n个元素被分成k类，每类的个数分别是n1,n2,...nk这n个元素的全排列数为 n!/(n1！×n2！×...×nk!). k类元素，每类的个数无限，从中取出m个元素的组合数为C(m+k-1,m）。
组合数学课件Lecture 4
16
Example
• Determine the permutation of {1, 2, 3, 4, 5, 6 ,7 ,8} whose inversion sequence is 5, 3, 4 ,0, 2, 1, 1, 0. • 8 1 • 87 2 1 • 867 2 1 3 • 8657 • 48657 4 2 1 3 • 486537 4 2 5 1 3 • 4862537 4 6 2 5 1 3 • 48625137
4 3 2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1
2 4 3 1 4 2 3 1 4 2 1 3 2 4 1 3 2 1 4 3 2 1 3 4
Chapter 4 Generating Permutations and Combinations
1
Summary
• • • • Generating permutations Inversions in Permutations Generating combinations Generating r-combinations
3
Example for mobile integer
• • For 2 6 3 1 5 4 only 3, 5, and 6 are mobile. Integer 1 can never be mobile since there is no integer smaller than 1. • The integer n is mobile, except in two cases: i) n is the first integer and its arrow points to the left. ii) n is the last integer and its arrow points to the right.
排列组合生成算法之字典序法和换位法动画ppt
2
例题：请写出1234的全排列
Step1：以1234开始排序，在每个数字上标一向 左箭头。 箭头所指一侧相邻的数若比它小，则称该数处于活 动状态 Step2：找最大的处于活动状态的数m Step3：将m与其箭头所指的邻数互换位置 Step4：将所得排列中比m大的数p的方向调整， 即改为相反方向 Step5：重复Step2~Step4直至无活动状态的数
2
例题：请写出1234的全排列 1 2 3 4
1234
2
例题：请写出1234的全排列 1 2 4 3
1234 1243
2
例题：请写出1234的全排列 1 4 2 3
1234 1243 1423
2
例题：请写出1234的全排列 4 1 2 3
1234 1243 1423 4123
2
例题：请写出1234的全排列 4 1 3 2
i-1者)
(4) 反排pj后面的数得到(q):
p1…pi-2 pj pnpj+1pi-1pj-1 ….pi+1 pi
1
例题：设有排列(p) =2763541, 按照字典式排 序, 它的下一个排列是谁?
Step1:求 i=maxj pj-1pj (找最后一个正序)
2
7
6
3
5
4
1
1
例题：设有排列(p) =2763541, 按照字典式排序, 它的下一个 排列是谁?
2
例题：请写出1234的全排列
Step1：以1234开始排序，在每个数字上标一向 左箭头。 箭头所指一侧相邻的数若比它小，则称该数处于活 动状态 Step2：找最大的处于活动状态的数m Step3：将m与其箭头所指的邻数互换位置 Step4：将所得排列中比m大的数p的方向调整， 即改为相反方向 Step5：重复Step2~Step4直至无活动状态的数
其它以字典的方式暴力破解压缩文件密码
其它以字典的⽅式暴⼒破解压缩⽂件密码背景 以前加密了⼀个压缩⽂件，⾥⾯全是回忆的照⽚，结果密码搞忘记了，我勒个去，然后就在⽹上下载了个软件暴⼒破解，结果很明显，没有破解出来。
然后⾃⼰就思索怎么破解。
⼀开始是加QQ群，让别⼈帮忙破解，仍然没有破解出来，别个还帮我跑了100g的字典，还是挺感谢他的，再说⼀声谢谢了哈，没有收我钱，全凭热⼼。
然后⾃⼰就寻思着，看怎么破解，暴⼒破解估计没戏，除⾮搭建云，⽤N多电脑协同破解，这⽅法对我⽽⾔不可取。
接着就想，还是字典破解吧，然后字典就很重要了撒，聪明的⿊客，不可能什么都暴⼒破解，肯定会根据⽤户信息，⽣成特定的密码，这样破解的概率是最⼤的。
这压缩⽂件的密码⼜是我⾃⼰设定的，那么根据我的习惯得出⾃⼰的密码应该还是不难，就在⽹上找⽣成密码的算法，其实就是个数学⾥⾯的排列组合，⽤C#实现。
然后就是把密码给搞出来，把常⽤密码拆分⼀下就可以了，根据⾃⼰的习惯来。
⽐如，我喜欢⽤guxingyue这个密码，然后我就拆分为gu、xing、yue，还有喜欢⽤guxingy这个密码，这密码是⼀个整体下⾯是我的原始字典经过排序组合，变成这个样⼦：解释⼀下哈，有1个密码的组合，有2个密码的组合，有3个密码的组合，有4个密码的组合排列。
总共就只有4个最基本的密码。
⽣成的密码数量就是这样多。
这样⽣成后的密码，就可以做为字典⽤于暴⼒破解了。
⽤到的代码：1、排列组合：public class PermutationAndCombination<T>{///<summary>///交换两个变量///</summary>///<param name="a">变量1</param>///<param name="b">变量2</param>public static void Swap(ref T a, ref T b){T temp = a;a = b;b = temp;}///<summary>///递归算法求数组的组合(私有成员)///</summary>///<param name="list">返回的范型</param>///<param name="t">所求数组</param>///<param name="n">辅助变量</param>///<param name="m">辅助变量</param>///<param name="b">辅助数组</param>for (int i = n; i >= m; i--){b[m - 1] = i - 1;if (m > 1){GetCombination(ref list, t, i - 1, m - 1, b, M);}else{if (list == null){list = new List<T[]>();}T[] temp = new T[M];for (int j = 0; j < b.Length; j++){temp[j] = t[b[j]];}list.Add(temp);}}}///<summary>///递归算法求排列(私有成员)///</summary>///<param name="list">返回的列表</param>///<param name="t">所求数组</param>///<param name="startIndex">起始标号</param>///<param name="endIndex">结束标号</param>private static void GetPermutation(ref List<T[]> list, T[] t, int startIndex, int endIndex) {if (startIndex == endIndex){if (list == null){list = new List<T[]>();}T[] temp = new T[t.Length];t.CopyTo(temp, 0);list.Add(temp);}else{for (int i = startIndex; i <= endIndex; i++){Swap(ref t[startIndex], ref t[i]);GetPermutation(ref list, t, startIndex + 1, endIndex);Swap(ref t[startIndex], ref t[i]);}}}///<summary>///求从起始标号到结束标号的排列，其余元素不变///</summary>///<param name="t">所求数组</param>///<param name="startIndex">起始标号</param>///<param name="endIndex">结束标号</param>///<returns>从起始标号到结束标号排列的范型</returns>public static List<T[]> GetPermutation(T[] t, int startIndex, int endIndex){if (startIndex < 0 || endIndex > t.Length - 1){return null;}List<T[]> list = new List<T[]>();GetPermutation(ref list, t, startIndex, endIndex);return list;}///<summary>///返回数组所有元素的全排列///</summary>///<param name="t">所求数组</param>///<returns>全排列的范型</returns>public static List<T[]> GetPermutation(T[] t){return GetPermutation(t, 0, t.Length - 1);}///<summary>///求数组中n个元素的排列///</summary>///<param name="t">所求数组</param>{if (n > t.Length){return null;}List<T[]> list = new List<T[]>();List<T[]> c = GetCombination(t, n);for (int i = 0; i < c.Count; i++){List<T[]> l = new List<T[]>();GetPermutation(ref l, c[i], 0, n - 1);list.AddRange(l);}return list;}///<summary>///求数组中n个元素的组合///</summary>///<param name="t">所求数组</param>///<param name="n">元素个数</param>///<returns>数组中n个元素的组合的范型</returns>public static List<T[]> GetCombination(T[] t, int n){if (t.Length < n){return null;}int[] temp = new int[n];List<T[]> list = new List<T[]>();GetCombination(ref list, t, t.Length, n, temp, n);return list;}}View Code2、⽣成密码：public void test4(){//排列组合⽣成密码StringBuilder sb = new StringBuilder();var strContent = System.IO.File.ReadAllText("02 pwd.txt");List<string> list = Regex.Split(strContent, "\r\n").ToList();//⼀个sheet包含的所有⾏，⾏与⾏之间⽤回车分隔//这⾥的1234，就是组合的个数new List<int> { 1, 2, 3, 4 }.ForEach(num =>{List<string[]> ListCombination = PermutationAndCombination<string>.GetPermutation(list.ToArray(), num); //求全部的3-3组合foreach (var arr in ListCombination){string pwd = "";//排列组合后的密码foreach (var item in arr){pwd += item;}sb.Append(pwd + "\r\n");}Console.WriteLine($"num:{num} count:{ListCombination.Count}");});System.IO.File.WriteAllText("222.txt", sb.ToString());}View Code⽂件解释：02 pwd.txt ⾥⾯存的是最基本的密码，就是⽤于组合排列的那⼏个基本元素，数学⾥⾯叫什么忘记了，突然感觉⾼中就像⽩学了⼀样，这都能忘！！还是没有经常⽤到的缘故吧222.txt ⾥⾯存的是排序组合后的密码⽤到的破解⼯具： 都是百度搜索的，这⾥鄙视⼀下百度，搜索经常是⼴告⼀连串，真是⽆聊。
组合推理 胡海霞
Key Words: permutations, permutations with repetition, combinations; combinatorial models; combinatorial operations; combinatorial reasoning
2006 届研究生硕士学位论文
学校代码：10269 学 号：YS03121002
影响高中生组合推理的因素
院
系： 理 工 学 院 数 学 系
专
业： 课 程 与 教 学 论
研 究 方 向： 数学史与数学教育
指 导 教 师： 汪 晓 勤 副 教 授
硕士研究生： 胡
海霞
2006 年 5 月完成
Dissertation of Master 2006
ABSTRACT
Permutation and combination, which are based on the two fundamental principles of counting, are two important methods of counting. As the way of thinking involved in permutation and combination is distinctive, the instruction of this topic can enhance students’ capabilities of logic thinking, mathematical modeling and reading and establishing proper conceptions of mathematics. Moreover, permutation and combination are the basis of probability. Therefore, they play an important role in senior middle school mathematics. However, it is difficult for the students to grasp and for the teacher to teach permutation and combination, and middle school students often make mistakes in combinatorial reasoning, which has also been revealed by several researches finished by foreign scholars. How to effectively improve on teaching of this topic is an important task facing the mathematics teachers in senior middle schools.
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E.g. 3 A corporation has a board of directors consisting of 10 members. The board must select from its members a chairperson, vice chairperson, and secretary. In how many ways c, 4)
E.g. 13 Group Picture The student council at Gotham College is made up of four freshmen, five sophomores, six juniors, and seven seniors. A yearbook photographer would like to line up three council members from each class for a picture. How many different pictures are possible if each group of classmates stands together?
E.g. 11 If you have six books, in how many ways can you select four books and arrange them on a shelf?
E.g. 12 Committee Selection The student council at City College has 12 students, of which 3 are seniors. How many committees of 4 students can be selected if at least one member of each committee must be a senior?
E.g. 4 In how many ways can a baseball team of nine players arrange themselves in a line for a group picture?
E.g. 5 A certain state uses automobile license plates that consist of three letters followed by three digits. How many such license plates are there?
Applying the Permutation and Combination Formula (b) P(6,4) (e) C(6,4) (c) P(5, 5) (f) C(5,5)
(a)P(100,2) (d) C(100,2)
E.g. 6 The board of directors of a corporation has 10 members. In how many ways can they choose a committee of three board members to negotiate a merger? E.g. 7 Eight horses are entered in a race in which a first, second and third prize will be awarded. Assuming no ties, how many different outcomes are possible?
Problem A How many words (by which we mean strings of letters) of two distinct letters can be formed from the letters {a, b, c}?
Problem B A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen?
E.g. 2 Counting Routs for a Trip An airline passenger must fly from New York to Frankfurt via London. There are 8 flights leaving New York for London. All of these provide connections on any one of 19 flights from London to Frankfurt. In how many different ways can the passenger book reservations?
[number of outcomes in E ] Pr( E ) N
C (8,3) 56 C (10,3) 120
Exercise: Balls in an Urn An urn contains seven green balls and five white balls. A sample of three balls is selected at random from the urn. Find the probability that (a) Only green balls are selected (b) At least one white ball is selected.
E = “all three balls selected are green” F = “at least one white ball is selected” The event E is the complement of F E= FC Pr(F)=1-Pr(FC)=1-Pr(E)
COMBINATION
AND
PERMUTATION FORMULA
THE MULTIPLICATION PRINCIPLE
E.g. 1 Counting Paths through a Maze A medical researcher wishes to test the effect of a drug on a rat’s perception by studying the rat’s ability to run a maze while under the influence of the drug. The maze is constructed so that to arrive at the exit point C, the rat must pass through a central point B. There are five paths from the entry point A to B, and three paths from B to C. In how many different ways can the rat run to the maze from A to C?
4! ∙P(4, 3) ∙P(5, 3) ∙P(6, 3) ∙P(7, 3)
E.g. 14 Displaying Paintings An art gallery has five paintings, by each of three artists, hanging in a row, with paintings by the same artist grouped together. How many different arrangements are possible?
E.g. 8 A political pollster wished to survey 1500 individuals chosen from a sample of 5,000,000 adults. In how many way s can the 1500 individuals be chosen?
E.g. 9 A club has 10 members. In how many ways can they choose a slate of four officers, consisting of a president, vice president, secretary, and treasurer? E.g. 10 Three couples go on a movie date. In how many ways can they be seated in a row of six seats so that each couple is seated together?
Multiplication Principle Suppose that a task is composed of two consecutive choices. If choice 1 can be performed in m ways and, for each of these, choice 2 can be performed in n ways, then the complete task can be performed in m∙n ways.
n n! n Cx x (n x) x!
P(n, 1)= n P(n, 1)= n(n-1) P(n, 1)= n(n-1) (n-2)
P(n, r)= n(n-1) (n-2)……(n-r+1)
P(n, r ) n(n 1) (n r 1) C (n, r ) r! r (r 1) 1
P(n, r)= the number of permutations of n objects taken r at a time C(n, r)= the number of combination of n objects taken r at a time