2021年高考數(shù)學(xué)真題和模擬題分類匯編專題13排列組合與二項(xiàng)式定理【含答案】

專題13排列組合與二項(xiàng)式定理/r/n一、選擇題部分/r/n1/r/n./r/n(2021?河南開(kāi)封三模/r/n?理/r/nT/r/n11/r/n)/r/n某校組織甲、乙兩個(gè)班的學(xué)生到“農(nóng)耕村”參加社會(huì)實(shí)踐活動(dòng)，某天安排有釀酒、油坊、陶藝、打鐵、紡織、竹編制作共六項(xiàng)活動(dòng)可供選擇，每個(gè)班上午、下午各安排一項(xiàng)活動(dòng)（不重復(fù)），且同一時(shí)間內(nèi)每項(xiàng)活動(dòng)都只允許一個(gè)班參加，則活動(dòng)安排方案的種數(shù)為（）/r/nA/r/n．/r/n126/r/n /r/nB/r/n．/r/n360/r/n /r/nC/r/n．/r/n600/r/n /r/nD/r/n．/r/n630/r/nD/r/n．/r/n第一類，上下午共安排/r/n4/r/n個(gè)活動(dòng)（上午/r/n2/r/n個(gè)，下午/r/n2/r/n個(gè)）分配給甲，乙，故有/r/nA/r/n6/r/n2/r/nA/r/n4/r/n2/r/n＝/r/n360/r/n種，/r/n第二類，上下午共安排/r/n3/r/n個(gè)活動(dòng)，（上午/r/n2/r/n個(gè)下午/r/n1/r/n個(gè)，或上午/r/n1/r/n個(gè)下午/r/n2/r/n個(gè)）分配給甲，乙，故有/r/nA/r/n6/r/n2/r/nA/r/n4/r/n1/r/nA/r/n2/r/n1/r/n＝/r/n240/r/n第三類，上下午共安排/r/n2/r/n個(gè)活動(dòng)，（上午/r/n1/r/n個(gè)，下午/r/n1/r/n個(gè)）分配給甲，乙，故有/r/nA/r/n6/r/n2/r/n＝/r/n30/r/n種，/r/n根據(jù)分類計(jì)數(shù)原理，共有/r/n360+240+30/r/n＝/r/n630/r/n種．/r/n2.(2021?河南開(kāi)封三模?理T/r/n6/r/n)/r/n（/r/na/r/n﹣/r/nx/r/n）（/r/n1+/r/nx/r/n）/r/n6/r/n的展開(kāi)式中/r/nx/r/n的奇數(shù)次冪項(xiàng)的系數(shù)之和為/r/n64/r/n，則實(shí)數(shù)/r/na/r/n＝（）/r/nA/r/n．/r/n4/r/n /r/nB/r/n．/r/n3/r/n /r/nC/r/n．/r/n2/r/n /r/nD/r/n．/r/n1/r/nB/r/n．/r/n∵（/r/na/r/n﹣/r/nx/r/n）（/r/n1+/r/nx/r/n）/r/n6/r/n的展開(kāi)式中/r/nx/r/n的奇數(shù)次冪項(xiàng)的系數(shù)之和為/r/n64/r/n，/r/n設(shè)/r/nf/r/n（/r/nx/r/n）＝（/r/na/r/n﹣/r/nx/r/n）（/r/nx/r/n+1/r/n）/r/n6/r/n＝/r/na/r/n0/r/n+/r/na/r/n1/r/nx/r/n+/r/na/r/n2/r/nx/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/nx/r/n6/r/n+/r/na/r/n6/r/nx/r/n7/r/n，/r/n令/r/nx/r/n＝/r/n1/r/n，則/r/nf/r/n（/r/n1/r/n）＝/r/na/r/n0/r/n+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/na/r/n3/r/n+/r/na/r/n4/r/n+/r/na/r/n5/r/n+/r/na/r/n6/r/n+/r/na/r/n7/r/n＝（/r/na/r/n+1/r/n）（/r/n1/r/n﹣/r/n1/r/n）/r/n5/r/n＝/r/n64/r/n（/r/na/r/n﹣/r/n1/r/n），/r/n①/r/n令/r/nx/r/n＝﹣/r/n1/r/n，則/r/nf/r/n（﹣/r/n1/r/n）＝/r/na/r/n0/r/n﹣/r/na/r/n1/r/n+/r/na/r/n2/r/n﹣/r/na/r/n3/r/n+/r/na/r/n4/r/n﹣/r/na/r/n5/r/n+/r/na/r/n6/r/n﹣/r/na/r/n7/r/n＝（/r/na/r/n﹣/r/n1/r/n）（﹣/r/n1/r/n﹣/r/n1/r/n）/r/n5/r/n＝/r/n0/r/n；/r/n②/r/n①/r/n﹣/r/n②/r/n得，/r/n2/r/n（/r/na/r/n1/r/n+/r/na/r/n3/r/n+/r/na/r/n5/r/n+/r/na/r/n7/r/n）＝/r/n64/r/n（/r/na/r/n﹣/r/n1/r/n），∴/r/na/r/n1/r/n+/r/na/r/n3/r/n+/r/na/r/n5/r/n+/r/na/r/n7/r/n＝/r/n32/r/n（/r/na/r/n﹣/r/n1/r/n）＝/r/n64/r/n，/r/n解得/r/na/r/n＝/r/n3/r/n．/r/n3.(2021?河南焦作三模?理T/r/n7/r/n)為/r/n了加強(qiáng)新型冠狀病毒疫情防控，某社區(qū)派遣甲、乙、丙、丁、戊五名志愿者參加/r/nA/r/n，/r/nB/r/n，/r/nC/r/n三個(gè)小區(qū)的防疫工作，每人只去/r/n1/r/n個(gè)小區(qū)，每個(gè)小區(qū)至少去/r/n1/r/n人，且甲、乙兩人約定去同一個(gè)小區(qū)，則不同的派遣方案共有（）/r/nA/r/n．/r/n24/r/n種/r/n /r/nB/r/n．/r/n36/r/n種/r/n /r/nC/r/n．/r/n48/r/n種/r/n /r/nD/r/n．/r/n64/r/n種/r/nB/r/n．/r/n根據(jù)題意，分/r/n2/r/n步進(jìn)行分析：/r/n①/r/n先將/r/n5/r/n人分成/r/n3/r/n組，要求甲乙在同一組，/r/n若甲乙兩人一組，將其他三人分成/r/n2/r/n組即可，有/r/nC/r/n3/r/n2/r/n種分組方法，/r/n若甲乙兩人與另外一人在同一組，有/r/nC/r/n3/r/n1/r/n種分組方法，/r/n則有/r/nC/r/n3/r/n1/r/n+/r/nC/r/n3/r/n2/r/n＝/r/n6/r/n種分組方法；/r/n②/r/n將分好的三組全排列，對(duì)應(yīng)/r/nA/r/n、/r/nB/r/n、/r/nC/r/n三個(gè)小區(qū)，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n種情況，/r/n則有/r/n6/r/n×/r/n6/r/n＝/r/n36/r/n種不同的派遣方案．/r/n4.(2021?河北張家口三模?T/r/n7/r/n)/r/n（/r/nx/r/n+2/r/ny/r/n﹣/r/n3/r/nz/r/n）/r/n5/r/n的展開(kāi)式中所有不含/r/ny/r/n的項(xiàng)的系數(shù)之和為（）/r/nA/r/n．﹣/r/n32/r/n /r/nB/r/n．﹣/r/n16/r/n /r/nC/r/n．/r/n10/r/n /r/nD/r/n．/r/n64/r/nA/r/n．/r/n在（/r/nx/r/n+2/r/ny/r/n﹣/r/n3/r/nz/r/n）/r/n7/r/n的展開(kāi)式中，通項(xiàng)公式為/r/n．/r/n若展開(kāi)式中的項(xiàng)不含/r/ny/r/n，則/r/nr/r/n＝/r/n6/r/n5/r/n展開(kāi)式中的所有項(xiàng)．/r/n令/r/nx/r/n＝/r/nz/r/n＝/r/n1/r/n，得這些項(xiàng)的系數(shù)之和為（﹣/r/n3/r/n）/r/n5/r/n＝﹣/r/n32/r/n．/r/n5.(2021?江西上饒三模?理T/r/n10/r/n．)現(xiàn)/r/n有語(yǔ)文、數(shù)學(xué)、英語(yǔ)、物理、化學(xué)、生物各/r/n1/r/n本書(shū)，把這/r/n6/r/n本書(shū)分別放入/r/n3/r/n個(gè)不同的抽屜里，要求每個(gè)抽屜至少放一本書(shū)且語(yǔ)文和數(shù)學(xué)放在同一個(gè)抽屜里，則不同的放法總數(shù)為（）/r/nA/r/n．/r/n78/r/n /r/nB/r/n．/r/n126/r/n /r/nC/r/n．/r/n148/r/n /r/nD/r/n．/r/n150/r/nD/r/n．/r/n根據(jù)題意，分/r/n2/r/n步進(jìn)行分析：/r/n①/r/n將/r/n6/r/n本書(shū)分為/r/n3/r/n組，要求語(yǔ)文和數(shù)學(xué)在同一組，/r/n若分為/r/n2/r/n、/r/n2/r/n、/r/n2/r/n的三組，有/r/nC/r/n4/r/n2/r/n＝/r/n3/r/n種分組方法，/r/n若分為/r/n3/r/n、/r/n2/r/n、/r/n1/r/n的三組，有/r/nC/r/n4/r/n1/r/n+/r/nC/r/n4/r/n1/r/nC/r/n3/r/n2/r/n＝/r/n16/r/n種分組方法，/r/n若分為/r/n4/r/n、/r/n1/r/n、/r/n1/r/n的三組，有/r/nC/r/n4/r/n2/r/n＝/r/n6/r/n種分組方法，/r/n則有/r/n3+16+6/r/n＝/r/n25/r/n種分組方法；/r/n②/r/n將分好三組全排列，放入三個(gè)不同的抽屜，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n種情況，/r/n則有/r/n25/r/n×/r/n6/r/n＝/r/n150/r/n種安排方法/r/n．/r/n6.(2021?重慶名校聯(lián)盟三模?T/r/n5/r/n．)/r/n已知（/r/n2/r/nx/r/n2/r/n+1/r/n）（/r/n﹣/r/n1/r/n）/r/n5/r/n的展開(kāi)式中各項(xiàng)系數(shù)之和為/r/n0/r/n，則該展開(kāi)式的常數(shù)項(xiàng)是（）/r/nA/r/n．﹣/r/n10/r/n /r/nB/r/n．﹣/r/n7/r/n /r/nC/r/n．/r/n9/r/n /r/nD/r/n．/r/n10/r/nC/r/n．/r/n（/r/n2/r/nx/r/n2/r/n+1/r/n）（/r/n）/r/n5/r/n開(kāi)式中各數(shù)和為/r/n3/r/n（/r/na/r/n﹣/r/n1/r/n）/r/n5/r/n＝/r/n0/r/n，∴/r/na/r/n＝/r/n1/r/n，/r/n則（/r/n）/r/n5/r/n，即/r/n，它的展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n?（﹣/r/n1/r/n）/r/nr/r/n?/r/nx/r/n2/r/nr/r/n﹣/r/n10/r/n，/r/n令/r/n2/r/nr/r/n﹣/r/n10/r/n＝﹣/r/n2/r/n，求得/r/nr/r/n＝/r/n4/r/n；令/r/n2/r/nr/r/n﹣/r/n10/r/n＝/r/n0/r/n，求得/r/nr/r/n＝/r/n5/r/n，/r/n故（/r/n2/r/nx/r/n2/r/n+1/r/n）（/r/n）/r/n5/r/n＝（/r/n2/r/nx/r/n2/r/n+1/r/n）（/r/n﹣/r/n1/r/n）/r/n5/r/n的展開(kāi)式中常數(shù)項(xiàng)是/r/n2/r/n﹣/r/n＝/r/n9/r/n．/r/n7.(2021?遼寧朝陽(yáng)三模?T/r/n8/r/n．)在/r/n三棱柱/r/nABC/r/n﹣/r/nA/r/n1/r/nB/r/n1/r/nC/r/n1/r/n中，/r/nD/r/n為側(cè)棱/r/nCC/r/n1/r/n的中點(diǎn)，從該三棱柱的九條棱中隨機(jī)選取兩條，則這兩條棱所在直線至少有一條與直線/r/nBD/r/n異面的概率是（）/r/nA/r/n．/r/n /r/nB/r/n．/r/n /r/nC/r/n．/r/n /r/nD/r/n．/r/nB/r/n．/r/n∵在三棱柱/r/nABC/r/n﹣/r/nA/r/n1/r/nB/r/n1/r/nC/r/n1/r/n中，/r/nD/r/n為側(cè)棱/r/nCC/r/n1/r/n的中點(diǎn)，/r/n∴該三棱柱的九條棱中與/r/nBD/r/n異面的棱有/r/n5/r/n條，/r/n從該三棱柱的九條棱中隨機(jī)選取兩條，/r/n基本事件總數(shù)/r/nn/r/n＝/r/n＝/r/n36/r/n，/r/n這兩條棱所在直線至少有一條與直線/r/nBD/r/n異面包含的基本事件個(gè)數(shù)為：/r/nm/r/n＝/r/n+/r/n＝/r/n26/r/n，/r/n則這兩條棱所在直線至少有一條與直線/r/nBD/r/n異面的概率/r/nP/r/n＝/r/n＝/r/n＝/r/n．/r/n8.(2021?遼寧朝陽(yáng)三模?T/r/n6/r/n．)下/r/n列各項(xiàng)中，是（/r/n）/r/n6/r/n展開(kāi)式中的項(xiàng)為（）/r/nA/r/n．/r/n15/r/n /r/nB/r/n．﹣/r/n20/r/nx/r/n2/r/n /r/nC/r/n．/r/n15/r/ny/r/n4/r/n /r/nD/r/n．﹣/r/n20/r/nC/r/n．/r/n（/r/n）/r/n6/r/n展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n＝（﹣/r/n1/r/n）/r/nr/r/n，/r/n由/r/n＝/r/n0/r/n且/r/n＝/r/n0/r/n，此時(shí)/r/nr/r/n無(wú)解，故展開(kāi)式中沒(méi)有常數(shù)項(xiàng)，故/r/nA/r/n錯(cuò)誤；/r/n由/r/n＝/r/n2/r/n且/r/n＝/r/n0/r/n，此時(shí)/r/nr/r/n無(wú)解，故展開(kāi)式中不含/r/nx/r/n2/r/n項(xiàng)，故/r/nB/r/n錯(cuò)誤；/r/n由/r/n＝/r/n0/r/n且/r/n＝/r/n4/r/n，此時(shí)/r/nr/r/n＝/r/n2/r/n，故/r/nT/r/n3/r/n＝（﹣/r/n1/r/n）/r/n2/r/ny/r/n4/r/n＝/r/n15/r/ny/r/n4/r/n，故/r/nC/r/n正確；/r/n由/r/n＝/r/n0/r/n且/r/n＝/r/n，此時(shí)/r/nr/r/n無(wú)解，故展開(kāi)式中不含/r/n項(xiàng)，故/r/nD/r/n錯(cuò)誤．/r/n9.(2021?湖南三模?T/r/n9/r/n．)在/r/n（/r/n3/r/nx/r/n﹣/r/n）/r/nn/r/n的展開(kāi)式中，各項(xiàng)系數(shù)和與二項(xiàng)式系數(shù)和之和為/r/n128/r/n，則（）/r/nA/r/n．二項(xiàng)式系數(shù)和為/r/n64/r/n /r/nB/r/n．各項(xiàng)系數(shù)和為/r/n64/r/n /r/nC/r/n．常數(shù)項(xiàng)為﹣/r/n135/r/n /r/nD/r/n．常數(shù)項(xiàng)為/r/n135/r/nABD/r/n．/r/n（/r/n3/r/nx/r/n﹣/r/n）/r/nn/r/n的展開(kāi)式中，各項(xiàng)系數(shù)和與二項(xiàng)式系數(shù)和之和為/r/n2/r/nn/r/n+2/r/nn/r/n＝/r/n128/r/n，∴/r/nn/r/n＝/r/n6/r/n，故二項(xiàng)式系數(shù)和為/r/n2/r/n6/r/n＝/r/n64/r/n，二項(xiàng)式系數(shù)和之和為/r/n2/r/nn/r/n＝/r/n2/r/n6/r/n＝/r/n64/r/n，故/r/nA/r/n、/r/nB/r/n正確；/r/n故展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n?（﹣/r/n1/r/n）/r/nr/r/n?/r/n3/r/n6/r/n﹣/r/nr/r/n?/r/n，令/r/n6/r/n﹣/r/n＝/r/n0/r/n，求得/r/nr/r/n＝/r/n4/r/n，/r/n故常數(shù)項(xiàng)為/r/n?/r/n3/r/n2/r/n＝/r/n135/r/n，故/r/nD/r/n正確．/r/n10.(2021?福建寧德三模?T7)/r/n《周髀算經(jīng)》是中國(guó)最古老的天文學(xué)、數(shù)學(xué)著作，公元/r/n3/r/n世紀(jì)初中國(guó)數(shù)學(xué)家趙爽創(chuàng)制了“勾股圓方圖”/r/n(/r/n如圖/r/n)/r/n，用以證明其中記載的勾股定理/r/n./r/n現(xiàn)提供/r/n4/r/n種不同顏色給右圖中/r/n5/r/n個(gè)區(qū)域涂色，規(guī)定每個(gè)區(qū)域只涂一種顏色，相鄰區(qū)域顏色不同，則不同涂色的方法種數(shù)為/r/n(?)/r/nA.36/r/n /r/nB.48/r/n /r/nC.72/r/n /r/nD.96/r/nC/r/n．/r/n根據(jù)題意，分/r/n2/r/n步進(jìn)行分析：/r/n

①對(duì)于區(qū)域/r/nABE/r/n，三個(gè)區(qū)域兩兩相鄰，有/r/nA/r/n4/r/n3/r/n=24/r/n種涂色的方法，/r/n

②對(duì)于區(qū)域/r/nCD/r/n，若/r/nC/r/n區(qū)域與/r/nA/r/n顏色相同，/r/nD/r/n區(qū)域有/r/n2/r/n種選法，/r/n

若/r/nC/r/n區(qū)域與/r/nA/r/n顏色不同，則/r/nC/r/n區(qū)域有/r/n1/r/n種選法，/r/nD/r/n區(qū)域也只有/r/n1/r/n種選法，/r/n

則區(qū)域/r/nCD/r/n有/r/n2+1=3/r/n種涂色的方法，/r/n

則有/r/n24×3=72/r/n種涂色的方法，/r/n

故選：/r/nC./r/n

/r/n根據(jù)題意，分/r/n2/r/n步依次分析區(qū)域/r/nABE/r/n和區(qū)域/r/nCD/r/n的涂色方法數(shù)目，由分步計(jì)數(shù)原理計(jì)算可得答案。/r/n

本題考查排列組合的應(yīng)用，涉及分步計(jì)數(shù)原理的應(yīng)用，屬于基礎(chǔ)題。/r/n

/r/n11.(2021?安徽宿州三模?理T/r/n6/r/n．)/r/n新冠肺炎疫情防控期間，按照宿州市疫情防控應(yīng)急指揮部的要求，市教育體育局對(duì)各市直學(xué)校下發(fā)了有關(guān)疫情防控通知．某學(xué)校按市局通知要求，制定了錯(cuò)峰放學(xué)，錯(cuò)峰吃飯的具體防疫措施．高三年級(jí)一層樓有/r/nA/r/n、/r/nB/r/n、/r/nC/r/n、/r/nD/r/n、/r/nE/r/n、/r/nF/r/n六個(gè)班排隊(duì)吃飯，/r/nA/r/n班必須排在第一位，且/r/nD/r/nA/r/n．/r/n20/r/n種/r/n /r/nB/r/n．/r/n56/r/n種/r/n /r/nC/r/n．/r/n72/r/n種/r/n /r/nD/r/n．/r/n40/r/n種/r/nC/r/n．/r/n根據(jù)題意，/r/nA/r/n班必須排在第一位，剩下/r/n5/r/n個(gè)班級(jí)安排在后面的/r/n5/r/n個(gè)位置即可，/r/n分/r/n2/r/n步進(jìn)行分析：/r/n①/r/n將/r/nBCF/r/n三個(gè)班級(jí)全排列，排好后有/r/n4/r/n個(gè)空位，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n種排法，/r/n②/r/n在/r/n4/r/n個(gè)空位中選出/r/n2/r/n個(gè)，安排/r/nD/r/n班、/r/nE/r/n班，有/r/nA/r/n42/r/n＝/r/n12/r/n種排法，/r/n則有/r/n6/r/n×/r/n12/r/n＝/r/n72/r/n種不同的方案/r/n．/r/n12.(2021?安徽宿州三模?理T/r/n7/r/n．)/r/n（/r/nx/r/n+2/r/ny/r/n）（/r/n2/r/nx/r/n+/r/ny/r/n）/r/n5/r/n的展開(kāi)式中/r/nx/r/n3/r/ny/r/n3/r/n項(xiàng)的系數(shù)為（）/r/nA/r/n．/r/n80/r/n /r/nB/r/n．/r/n160/r/n /r/nC/r/n．/r/n200/r/n /r/nD/r/n．/r/n240/r/nC/r/n．/r/n（/r/nx/r/n+2/r/ny/r/n）（/r/n2/r/nx/r/n+/r/ny/r/n）/r/n5/r/n＝（/r/nx/r/n+2/r/ny/r/n）（/r/n（/r/n2/r/nx/r/n）/r/n5/r/n+/r/n?（/r/n2/r/nx/r/n）/r/n4/r/ny/r/n+/r/n?（/r/n2/r/nx/r/n）/r/n3/r/ny/r/n2/r/n+/r/n?（/r/n2/r/nx/r/n）/r/n2/r/ny/r/n3/r/n+/r/n?（/r/n2/r/nx/r/n）?/r/ny/r/n4/r/n+/r/n?/r/ny/r/n5/r/n），/r/n故展開(kāi)式中/r/nx/r/n3/r/ny/r/n3/r/n項(xiàng)的系數(shù)為/r/n?/r/n2/r/n2/r/n+2/r/n?/r/n2/r/n3/r/n＝/r/n200/r/n．/r/n13.(2021?安徽馬鞍山三模?理T/r/n6/r/n．)/r/n的常數(shù)項(xiàng)為/r/n25/r/n，則實(shí)數(shù)/r/na/r/n的值為（）/r/nA/r/n．/r/n1/r/n /r/nB/r/n．﹣/r/n1/r/n /r/nC/r/n．/r/n2/r/n /r/nD/r/n．﹣/r/n2/r/nD/r/n．/r/n（/r/n2/r/nx/r/n﹣/r/n1/r/n）/r/n6/r/n的展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝（﹣/r/n1/r/n）/r/nr/r/n?/r/n2/r/n6/r/n﹣/r/nr/r/n?/r/n?/r/nx/r/n6/r/n﹣/r/nr/r/n，/r/n所以/r/n的常數(shù)項(xiàng)為（﹣/r/n1/r/n）/r/n5/r/n?/r/n2/r/n?/r/na/r/n+/r/n（﹣/r/n1/r/n）/r/n6/r/n?/r/n2/r/n0/r/n＝﹣/r/n12/r/na/r/n+1/r/n＝/r/n25/r/n，/r/n解得/r/na/r/n＝﹣/r/n2/r/n．/r/n1/r/n4/r/n.(2021/r/n?河北邯鄲二模?理/r/nT6/r/n．/r/n)/r/n（/r/nx/r/n2/r/n﹣/r/nx/r/n）（/r/n1+/r/nx/r/n）/r/n6/r/n的展開(kāi)式中/r/nx/r/n3/r/n項(xiàng)的系數(shù)為（）/r/nA/r/n．﹣/r/n9/r/n /r/nB/r/n．/r/n9/r/n /r/nC/r/n．﹣/r/n21/r/n /r/nD/r/n．/r/n21/r/nA/r/n．/r/n因?yàn)楦鶕?jù)（/r/n1+/r/nx/r/n）/r/n6/r/n展開(kāi)式的通項(xiàng)/r/nT/r/nr/r/n+1/r/n＝/r/n，/r/n所以（/r/nx/r/n2/r/n﹣/r/nx/r/n）（/r/n1+/r/nx/r/n）/r/n6/r/n的展開(kāi)式中/r/nx/r/n3/r/n項(xiàng)為/r/n＝﹣/r/n9/r/nx/r/n3/r/n，/r/n所以含/r/nx/r/n3/r/n項(xiàng)的系數(shù)﹣/r/n9/r/n．/r/n15.(2021?河北秦皇島二模?理T/r/n10/r/n．)/r/n已知（/r/n2/r/n﹣/r/nx/r/n）/r/n6/r/n＝/r/na/r/n0/r/n+/r/na/r/n1/r/nx/r/n+/r/na/r/n2/r/nx/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/nx/r/n6/r/n，則下列選項(xiàng)正確的是（）/r/nA/r/n．/r/na/r/n3/r/n＝﹣/r/n360/r/n /r/nB/r/n．（/r/na/r/n0/r/n+/r/na/r/n2/r/n+/r/na/r/n4/r/n+/r/na/r/n6/r/n）/r/n2/r/n﹣（/r/na/r/n1/r/n+/r/na/r/n3/r/n+/r/na/r/n5/r/n）/r/n2/r/n＝/r/n1/r/n /r/nC/r/n．/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/n＝（/r/n2/r/n﹣/r/n）/r/n6/r/n /r/nD/r/n．展開(kāi)式中系數(shù)最大的為/r/na/r/n2/r/nBD/r/n．/r/n令/r/nx/r/n＝/r/n0/r/n得/r/na/r/n0/r/n＝/r/n2/r/n6/r/n，/r/n令/r/nx/r/n＝/r/n1/r/n得/r/na/r/n0/r/n+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/n＝（/r/n2/r/n﹣/r/n）/r/n6/r/n，/r/n則/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/n＝（/r/n2/r/n﹣/r/n）/r/n6/r/n﹣/r/n2/r/n6/r/n，故/r/nC/r/n錯(cuò)誤，/r/n令/r/nx/r/n＝﹣/r/n1/r/n，得/r/na/r/n0/r/n﹣/r/na/r/n1/r/n+/r/na/r/n2/r/n﹣/r/na/r/n3/r/n+/r/na/r/n4/r/n﹣/r/na/r/n5/r/n+/r/na/r/n6/r/n＝（/r/n2+/r/n）/r/n6/r/n，/r/n則（/r/na/r/n0/r/n+/r/na/r/n2/r/n+/r/na/r/n4/r/n+/r/na/r/n6/r/n）/r/n2/r/n﹣（/r/na/r/n1/r/n+/r/na/r/n3/r/n+/r/na/r/n5/r/n）/r/n2/r/n＝（/r/na/r/n0/r/n+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/n）（/r/na/r/n0/r/n﹣/r/na/r/n1/r/n+/r/na/r/n2/r/n﹣/r/na/r/n3/r/n+/r/na/r/n4/r/n﹣/r/na/r/n5/r/n+/r/na/r/n6/r/n）＝（/r/n2/r/n﹣/r/n）/r/n6/r/n（/r/n2+/r/n）/r/n6/r/n＝/r/n[/r/n（/r/n2/r/n﹣/r/n）（/r/n2+/r/n）/r/n]/r/n6/r/n＝/r/n1/r/n，故/r/nB/r/n正確，/r/na/r/n3/r/n＝/r/n2/r/n3/r/n（/r/n）/r/n3/r/n＝﹣/r/n480/r/n，故/r/nA/r/n錯(cuò)誤，/r/n展開(kāi)式中偶數(shù)項(xiàng)系數(shù)為負(fù)值，奇數(shù)項(xiàng)系數(shù)為正值，/r/n則系數(shù)最大的在/r/na/r/n0/r/n，/r/na/r/n2/r/n，/r/na/r/n4/r/n，/r/na/r/n6/r/n中，/r/n展開(kāi)式的通項(xiàng)公式/r/nT/r/nk/r/n+1/r/n＝/r/n2/r/n6/r/n﹣/r/nk/r/n（/r/nx/r/n）/r/nk/r/n，/r/n則/r/na/r/n2/r/n＝/r/n720/r/n，/r/na/r/n4/r/n＝/r/n540/r/n，/r/na/r/n6/r/n＝/r/n27/r/n，/r/n則系數(shù)最大的為/r/na/r/n2/r/n．/r/n16.(2021?遼寧朝陽(yáng)二模?T6．)/r/n今年我國(guó)中醫(yī)藥選出的“三藥三方”對(duì)治療新冠肺炎均有顯著效果，功不可沒(méi)．“三藥”分別為金花清感顆粒、連花清瘟膠囊、血必清注射液；“三方”分別為清肺排毒湯、化濕敗毒方、宜肺敗毒方，若某醫(yī)生從“三藥三方”中隨機(jī)選出/r/n2/r/n種，則恰好選出/r/n1/r/n藥/r/n1/r/n方的方法種數(shù)為（）/r/nA/r/n．/r/n15/r/n /r/nB/r/n．/r/n30/r/n /r/nC/r/n．/r/n6/r/n /r/nD/r/n．/r/n9/r/nD/r/n．/r/n根據(jù)題意，某醫(yī)生從“三藥三方”中隨機(jī)選出/r/n2/r/n種，恰好選出/r/n1/r/n藥/r/n1/r/n方，/r/n則/r/n1/r/n藥的取法有/r/n3/r/n種，/r/n1/r/n方的取法也有/r/n3/r/n種，/r/n則恰好選出/r/n1/r/n藥/r/n1/r/n方的方法種數(shù)為/r/n3/r/n×/r/n3/r/n＝/r/n9/r/n．/r/n17.(2021?山東濰坊二模?T/r/n4/r/n．)/r/n2021/r/n年是中國(guó)共產(chǎn)黨百年華誕．某學(xué)校社團(tuán)將舉辦慶祝中國(guó)共產(chǎn)黨成立/r/n100/r/n周年革命歌曲展演．現(xiàn)從《歌唱祖國(guó)》《英雄贊歌》《唱支山歌給黨聽(tīng)》《毛主席派人來(lái)》/r/n4/r/n首獨(dú)唱歌曲和《沒(méi)有共產(chǎn)黨就沒(méi)有新中國(guó)》《我和我的祖國(guó)》/r/n2/r/n首合唱歌曲中共選出/r/n4/r/n首歌曲安排演出，要求最后一首歌曲必須是合唱，則不同的安排方法共有（）/r/nA/r/n．/r/n14/r/n /r/nB/r/n．/r/n48/r/n /r/nC/r/n．/r/n72/r/n /r/nD/r/n．/r/n120/r/nD/r/n．/r/n根據(jù)題意，在/r/n2/r/n首合唱歌曲中任選/r/n1/r/n首，安排在最后，有/r/n2/r/n種安排方法，/r/n在其他/r/n5/r/n首歌曲中任選/r/n3/r/n首，作為前/r/n3/r/n首歌曲，有/r/nA/r/n5/r/n3/r/n＝/r/n60/r/n種安排方法，/r/n則有/r/n2/r/n×/r/n60/r/n＝/r/n120/r/n種不同的安排方法/r/n．/r/n1/r/n8/r/n.(2021/r/n?浙江麗水湖州衢州二模?/r/nT8/r/n．/r/n)/r/n某/r/n市抽調(diào)/r/n5/r/n位醫(yī)生分赴/r/n4/r/n所醫(yī)院支援抗疫，要求每位醫(yī)生只能去一所醫(yī)院，每所醫(yī)院至少安排一位醫(yī)生．由于工作需要，甲、乙兩位醫(yī)生必須安排在不同的醫(yī)院，則不同的安排種數(shù)是（）/r/nA/r/n．/r/n90/r/n /r/nB/r/n．/r/n216/r/n /r/nC/r/n．/r/n144/r/n /r/nD/r/n．/r/n240/r/nB/r/n．/r/n根據(jù)題意，分/r/n2/r/n步進(jìn)行分析：/r/n①/r/n將/r/n5/r/n位醫(yī)生分為/r/n4/r/n組，要求甲乙不在同一組，有/r/nC/r/n5/r/n2/r/n﹣/r/n1/r/n＝/r/n9/r/n種分組方法，/r/n②/r/n將分好的/r/n4/r/n組安排到/r/n4/r/n所醫(yī)院支援抗疫，有/r/nA/r/n4/r/n4/r/n＝/r/n24/r/n種安排方法，/r/n則有/r/n9/r/n×/r/n24/r/n＝/r/n216/r/n種安排種數(shù)．/r/n二、填空題部分/r/n1/r/n9/r/n.(2021/r/n?山東濰坊二模?/r/nT13/r/n．/r/n)/r/n設(shè)/r/n（/r/nx/r/n+1/r/n）/r/n4/r/n＝/r/na/r/n0/r/n+/r/na/r/n1/r/nx/r/n+/r/na/r/n2/r/nx/r/n2/r/n+/r/na/r/n3/r/nx/r/n3/r/n+/r/na/r/n4/r/nx/r/n4/r/n，則/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/na/r/n3/r/n+/r/na/r/n4/r/n＝/r/n/r/n．/r/n15/r/n．/r/n∵（/r/nx/r/n+1/r/n）/r/n4/r/n＝/r/na/r/n0/r/n+/r/na/r/n1/r/nx/r/n+/r/na/r/n2/r/nx/r/n2/r/n+/r/na/r/n3/r/nx/r/n3/r/n+/r/na/r/n4/r/nx/r/n4/r/n，/r/n令/r/nx/r/n＝/r/n1/r/n得：/r/n2/r/n4/r/n＝/r/n16/r/n＝/r/na/r/n0/r/n+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/na/r/n3/r/n+/r/na/r/n4/r/n，/r/n令/r/nx/r/n＝/r/n0/r/n得：/r/n1/r/n＝/r/na/r/n0/r/n，∴/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/na/r/n3/r/n+/r/na/r/n4/r/n＝/r/n16/r/n﹣/r/n1/r/n＝/r/n15/r/n．/r/n20.(2021?遼寧朝陽(yáng)二模?T13．)在/r/n（/r/nx/r/n﹣/r/n2/r/ny/r/n+/r/nz/r/n）/r/n7/r/n的展開(kāi)式中，所有形如/r/nx/r/na/r/ny/r/nb/r/nz/r/n2/r/n（/r/na/r/n，/r/nb/r/n∈/r/nN/r/n）的項(xiàng)的系數(shù)之和是/r/n/r/n．/r/n﹣/r/n21/r/n．/r/n因?yàn)椋?r/nx/r/n﹣/r/n2/r/ny/r/n+/r/nz/r/n）/r/n7/r/n＝/r/n[/r/n（/r/nx/r/n﹣/r/n2/r/ny/r/n）/r/n+/r/nz/r/n]/r/n7/r/n，/r/n所以展開(kāi)式中含/r/nz/r/n2/r/n的項(xiàng)為/r/nC/r/n，/r/n令/r/nx/r/n＝/r/ny/r/n＝/r/nz/r/n＝/r/n1/r/n，則所求系數(shù)之和為/r/nC/r/n?（/r/n1/r/n﹣/r/n2/r/n）/r/n5/r/n?/r/n1/r/n2/r/n＝﹣/r/n21/r/n．/r/n2/r/n1.(2021/r/n?廣東潮州二模?/r/nT13/r/n．/r/n)/r/n（/r/nx/r/n﹣/r/n）/r/n6/r/n的展開(kāi)式的常數(shù)項(xiàng)是/r/n/r/n．用數(shù)字作答）．/r/n﹣/r/n20/r/n．/r/n（/r/nx/r/n﹣/r/n）/r/n6/r/n的展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n?（﹣/r/n1/r/n）/r/nr/r/n?/r/nx/r/n6/r/n﹣/r/n2/r/nr/r/n，/r/n令/r/n6/r/n﹣/r/n2/r/nr/r/n＝/r/n0/r/n，求得/r/nr/r/n＝/r/n3/r/n，可得展開(kāi)式的常數(shù)項(xiàng)為/r/n?（﹣/r/n1/r/n）＝﹣/r/n20/r/n．/r/n22/r/n.(2021/r/n?廣東潮州二模?/r/nT15/r/n．/r/n)/r/n根/r/n據(jù)中央關(guān)于精準(zhǔn)脫貧的要求，我市農(nóng)業(yè)經(jīng)濟(jì)部門(mén)隨機(jī)派遣/r/n4/r/n位專家對(duì)/r/n3/r/n個(gè)縣區(qū)進(jìn)行調(diào)研，每個(gè)縣區(qū)至少派/r/n1/r/n位專家，則專家派遣的方法的種數(shù)為/r/n/r/n．/r/n36/r/n．/r/n根據(jù)題意，分/r/n2/r/n步進(jìn)行分析：/r/n①/r/n將/r/n4/r/n位專家分為/r/n3/r/n組，有/r/nC/r/n4/r/n2/r/n＝/r/n6/r/n種分組方法，/r/n②/r/n將分好的三組全排列，分到/r/n3/r/n個(gè)縣區(qū)進(jìn)行調(diào)研，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n種情況，/r/n則有/r/n6/r/n×/r/n6/r/n＝/r/n36/r/n種派遣方法/r/n．/r/n23.(2021?天津南開(kāi)二模?T/r/n11/r/n．)/r/n的二項(xiàng)展開(kāi)式中，/r/nx/r/n3/r/n的系數(shù)等于/r/n/r/n．/r/n15/r/n．/r/n的二項(xiàng)展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n＝（﹣/r/n1/r/n）/r/nr/r/n，令/r/n6/r/n﹣/r/nr/r/n＝/r/n3/r/n且/r/n，解得/r/nr/r/n＝/r/n2/r/n，所以/r/nx/r/n3/r/n的系數(shù)等于（﹣/r/n5/r/n）/r/n2/r/n＝/r/n15/r/n．/r/n24.(2021?浙江卷?T13)/r/n已知多項(xiàng)式/r/n，則/r/n___________，/r/n___________/r/n./r/n(1)./r/n;/r/n(2)./r/n./r/n，/r/n，/r/n所以/r/n，/r/n，/r/n所以/r/n./r/n故答案為./r/n25.(2021?江蘇鹽城三模?T/r/n13/r/n)/r/n文旅部在/r/n2021/r/n年圍繞/r/n“/r/n重溫紅色歷史、傳承奮斗精神/r/n”“/r/n走進(jìn)大國(guó)重器、感受中國(guó)力量/r/n”“/r/n體驗(yàn)美麗鄉(xiāng)村、助力鄉(xiāng)村振興/r/n”/r/n三個(gè)主題，遴選推出/r/n“/r/n建黨百年紅色旅游百條精品線路/r/n”/r/n．這些精品線路中包含上海/r/n—/r/n大會(huì)址、嘉興南湖、井岡山、延安、西柏坡等/r/n5/r/n個(gè)傳統(tǒng)紅色旅游景區(qū)，還有港珠澳大橋、北京大興國(guó)際機(jī)場(chǎng)、/r/n“/r/n中國(guó)天眼/r/n”/r/n、/r/n“/r/n兩彈一星/r/n”/r/n紀(jì)念館、湖南十八洞村、浙江余村、貴州華茂村等/r/n7/r/n個(gè)展現(xiàn)改革開(kāi)放和新時(shí)代發(fā)展成就、展示科技強(qiáng)國(guó)和脫貧攻堅(jiān)成果的景區(qū)．為安排旅游路線，從上述/r/n12/r/n個(gè)景區(qū)中選/r/n3/r/n個(gè)景區(qū)，則至少含有/r/n1/r/n個(gè)傳統(tǒng)紅色旅游景區(qū)的選法有種．/r/n185/r/n．/r/n【考點(diǎn)】排列組合/r/n由題意，可用間接法，總體情況為從/r/n12/r/n個(gè)景區(qū)中選/r/n3/r/n個(gè)景區(qū)/r/nEQC/r/n\/r/no/r/n\/r/nal/r/n(\/r/nS/r/n\/r/nUP/r/n5(3),\/r/nS/r/n\/r/nDO/r/n3(12）/r/n，從/r/n7/r/n個(gè)非傳統(tǒng)紅色旅游景區(qū)中選/r/n3/r/n個(gè)景區(qū)/r/nEQC/r/n\/r/no/r/n\/r/nal/r/n(\/r/nS/r/n\/r/nUP/r/n5(3),\/r/nS/r/n\/r/nDO/r/n3(7）/r/n，則至少含有/r/n1/r/n個(gè)傳統(tǒng)紅色旅游景區(qū)的選法有/r/nEQC/r/n\/r/no/r/n\/r/nal/r/n(\/r/nS/r/n\/r/nUP/r/n5(3),\/r/nS/r/n\/r/nDO/r/n3(12）/r/n－/r/nEQC/r/n\/r/no/r/n\/r/nal/r/n(\/r/nS/r/n\/r/nUP/r/n5(3),\/r/nS/r/n\/r/nDO/r/n3(7）/r/n＝/r/n185/r/n．/r/n26.(2021?河南鄭州三模?理T/r/n14/r/n)/r/n展開(kāi)式中的/r/na/r/n與/r/nb/r/n指數(shù)相同的項(xiàng)的表達(dá)式為/r/n/r/n．/r/n﹣/r/n84/r/na/r/n．/r/n展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/nC/r/n＝/r/nC/r/n，令/r/n6/r/n﹣/r/n，解得/r/nr/r/n＝/r/n3/r/n，/r/n所以展開(kāi)式中的/r/na/r/n與/r/nb/r/n指數(shù)相同的項(xiàng)的表達(dá)式為/r/nC/r/n＝﹣/r/n84/r/na/r/n．/r/n27.(2021?四川內(nèi)江三模?理T/r/n14/r/n．)/r/n二項(xiàng)式（/r/n2/r/nx/r/n2/r/n﹣/r/n）/r/n6/r/n的展開(kāi)式中的常數(shù)項(xiàng)是/r/n/r/n．（用數(shù)字作答）/r/n60/r/n．/r/n（/r/n2/r/nx/r/n2/r/n﹣/r/n）/r/n6/r/n的展開(kāi)式的通項(xiàng)公式為/r/nT/r/nr/r/n+1/r/n＝/r/n?（﹣/r/n1/r/n）/r/nr/r/n?/r/n2/r/n7/r/n﹣/r/nr/r/n?/r/nx/r/n12/r/n﹣/r/n3/r/nr/r/n，/r/n令/r/n12/r/n﹣/r/n3/r/nr/r/n＝/r/n6/r/n，求得/r/nr/r/n＝/r/n4/r/n?/r/n2/r/n2/r/n＝/r/n60/r/n．/r/n28.(2021?上海嘉定三模?T/r/n8/r/n．)/r/n展開(kāi)式中的常數(shù)項(xiàng)為/r/n﹣/r/n19/r/n/r/n．/r/n﹣/r/n19/r/n．/r/n依題意，（/r/n1/r/n﹣/r/nx/r/n）/r/n6/r/n展開(kāi)式的通項(xiàng)是/r/n，/r/n當(dāng)/r/nr/r/n＝/r/n0/r/n時(shí)，/r/n；/r/n當(dāng)/r/nr/r/n＝/r/n3/r/n時(shí)，/r/n，/r/n∴/r/n展開(kāi)式的常數(shù)項(xiàng)是/r/n．/r/n29.(2021?四川瀘州三模?理T/r/n13/r/n．)/r/n（/r/nx/r/n+1/r/n）（/r/nx/r/n﹣/r/n1/r/n）/r/n6/r/n展開(kāi)式中/r/nx/r/n3/r/n項(xiàng)的系數(shù)為/r/n/r/n．/r/n﹣/r/n5/r/n．/r/n由題意可得展開(kāi)式中含/r/nx/r/n3/r/n項(xiàng)為/r/nx/r/n+1/r/n＝（/r/n15/r/n﹣/r/n20/r/n）/r/nx/r/n3/r/n＝﹣/r/n5/r/nx/r/n3/r/n．/r/n30.(2021?江蘇常數(shù)三模?T/r/n13/r/n．)/r/n的展開(kāi)式中常數(shù)項(xiàng)為/r/n/r/n．/r/n﹣/r/