2021年高考数学真题和模拟题分类汇编专题13排列组合与二项式定理【含答案】

专题13排列组合与二项式定理/r/n一、选择题部分/r/n1/r/n./r/n(2021•河南开封三模/r/n•理/r/nT/r/n11/r/n)/r/n某校组织甲、乙两个班的学生到“农耕村”参加社会实践活动，某天安排有酿酒、油坊、陶艺、打铁、纺织、竹编制作共六项活动可供选择，每个班上午、下午各安排一项活动（不重复），且同一时间内每项活动都只允许一个班参加，则活动安排方案的种数为（）/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第一类，上下午共安排/r/n4/r/n个活动（上午/r/n2/r/n个，下午/r/n2/r/n个）分配给甲，乙，故有/r/nA/r/n6/r/n2/r/nA/r/n4/r/n2/r/n＝/r/n360/r/n种，/r/n第二类，上下午共安排/r/n3/r/n个活动，（上午/r/n2/r/n个下午/r/n1/r/n个，或上午/r/n1/r/n个下午/r/n2/r/n个）分配给甲，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第三类，上下午共安排/r/n2/r/n个活动，（上午/r/n1/r/n个，下午/r/n1/r/n个）分配给甲，乙，故有/r/nA/r/n6/r/n2/r/n＝/r/n30/r/n种，/r/n根据分类计数原理，共有/r/n360+240+30/r/n＝/r/n630/r/n种．/r/n2.(2021•河南开封三模•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的展开式中/r/nx/r/n的奇数次幂项的系数之和为/r/n64/r/n，则实数/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的展开式中/r/nx/r/n的奇数次幂项的系数之和为/r/n64/r/n，/r/n设/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，则/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，则/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)为/r/n了加强新型冠状病毒疫情防控，某社区派遣甲、乙、丙、丁、戊五名志愿者参加/r/nA/r/n，/r/nB/r/n，/r/nC/r/n三个小区的防疫工作，每人只去/r/n1/r/n个小区，每个小区至少去/r/n1/r/n人，且甲、乙两人约定去同一个小区，则不同的派遣方案共有（）/r/nA/r/n．/r/n24/r/n种/r/n /r/nB/r/n．/r/n36/r/n种/r/n /r/nC/r/n．/r/n48/r/n种/r/n /r/nD/r/n．/r/n64/r/n种/r/nB/r/n．/r/n根据题意，分/r/n2/r/n步进行分析：/r/n①/r/n先将/r/n5/r/n人分成/r/n3/r/n组，要求甲乙在同一组，/r/n若甲乙两人一组，将其他三人分成/r/n2/r/n组即可，有/r/nC/r/n3/r/n2/r/n种分组方法，/r/n若甲乙两人与另外一人在同一组，有/r/nC/r/n3/r/n1/r/n种分组方法，/r/n则有/r/nC/r/n3/r/n1/r/n+/r/nC/r/n3/r/n2/r/n＝/r/n6/r/n种分组方法；/r/n②/r/n将分好的三组全排列，对应/r/nA/r/n、/r/nB/r/n、/r/nC/r/n三个小区，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n种情况，/r/n则有/r/n6/r/n×/r/n6/r/n＝/r/n36/r/n种不同的派遣方案．/r/n4.(2021•河北张家口三模•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的展开式中所有不含/r/ny/r/n的项的系数之和为（）/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的展开式中，通项公式为/r/n．/r/n若展开式中的项不含/r/ny/r/n，则/r/nr/r/n＝/r/n6/r/n5/r/n展开式中的所有项．/r/n令/r/nx/r/n＝/r/nz/r/n＝/r/n1/r/n，得这些项的系数之和为（﹣/r/n3/r/n）/r/n5/r/n＝﹣/r/n32/r/n．/r/n5.(2021•江西上饶三模•理T/r/n10/r/n．)现/r/n有语文、数学、英语、物理、化学、生物各/r/n1/r/n本书，把这/r/n6/r/n本书分别放入/r/n3/r/n个不同的抽屉里，要求每个抽屉至少放一本书且语文和数学放在同一个抽屉里，则不同的放法总数为（）/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根据题意，分/r/n2/r/n步进行分析：/r/n①/r/n将/r/n6/r/n本书分为/r/n3/r/n组，要求语文和数学在同一组，/r/n若分为/r/n2/r/n、/r/n2/r/n、/r/n2/r/n的三组，有/r/nC/r/n4/r/n2/r/n＝/r/n3/r/n种分组方法，/r/n若分为/r/n3/r/n、/r/n2/r/n、/r/n1/r/n的三组，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种分组方法，/r/n若分为/r/n4/r/n、/r/n1/r/n、/r/n1/r/n的三组，有/r/nC/r/n4/r/n2/r/n＝/r/n6/r/n种分组方法，/r/n则有/r/n3+16+6/r/n＝/r/n25/r/n种分组方法；/r/n②/r/n将分好三组全排列，放入三个不同的抽屉，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n种情况，/r/n则有/r/n25/r/n×/r/n6/r/n＝/r/n150/r/n种安排方法/r/n．/r/n6.(2021•重庆名校联盟三模•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的展开式中各项系数之和为/r/n0/r/n，则该展开式的常数项是（）/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开式中各数和为/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则（/r/n）/r/n5/r/n，即/r/n，它的展开式的通项公式为/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的展开式中常数项是/r/n2/r/n﹣/r/n＝/r/n9/r/n．/r/n7.(2021•辽宁朝阳三模•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为侧棱/r/nCC/r/n1/r/n的中点，从该三棱柱的九条棱中随机选取两条，则这两条棱所在直线至少有一条与直线/r/nBD/r/n异面的概率是（）/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为侧棱/r/nCC/r/n1/r/n的中点，/r/n∴该三棱柱的九条棱中与/r/nBD/r/n异面的棱有/r/n5/r/n条，/r/n从该三棱柱的九条棱中随机选取两条，/r/n基本事件总数/r/nn/r/n＝/r/n＝/r/n36/r/n，/r/n这两条棱所在直线至少有一条与直线/r/nBD/r/n异面包含的基本事件个数为：/r/nm/r/n＝/r/n+/r/n＝/r/n26/r/n，/r/n则这两条棱所在直线至少有一条与直线/r/nBD/r/n异面的概率/r/nP/r/n＝/r/n＝/r/n＝/r/n．/r/n8.(2021•辽宁朝阳三模•T/r/n6/r/n．)下/r/n列各项中，是（/r/n）/r/n6/r/n展开式中的项为（）/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展开式的通项公式为/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，此时/r/nr/r/n无解，故展开式中没有常数项，故/r/nA/r/n错误；/r/n由/r/n＝/r/n2/r/n且/r/n＝/r/n0/r/n，此时/r/nr/r/n无解，故展开式中不含/r/nx/r/n2/r/n项，故/r/nB/r/n错误；/r/n由/r/n＝/r/n0/r/n且/r/n＝/r/n4/r/n，此时/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正确；/r/n由/r/n＝/r/n0/r/n且/r/n＝/r/n，此时/r/nr/r/n无解，故展开式中不含/r/n项，故/r/nD/r/n错误．/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的展开式中，各项系数和与二项式系数和之和为/r/n128/r/n，则（）/r/nA/r/n．二项式系数和为/r/n64/r/n /r/nB/r/n．各项系数和为/r/n64/r/n /r/nC/r/n．常数项为﹣/r/n135/r/n /r/nD/r/n．常数项为/r/n135/r/nABD/r/n．/r/n（/r/n3/r/nx/r/n﹣/r/n）/r/nn/r/n的展开式中，各项系数和与二项式系数和之和为/r/n2/r/nn/r/n+2/r/nn/r/n＝/r/n128/r/n，∴/r/nn/r/n＝/r/n6/r/n，故二项式系数和为/r/n2/r/n6/r/n＝/r/n64/r/n，二项式系数和之和为/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正确；/r/n故展开式的通项公式为/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故常数项为/r/n•/r/n3/r/n2/r/n＝/r/n135/r/n，故/r/nD/r/n正确．/r/n10.(2021•福建宁德三模•T7)/r/n《周髀算经》是中国最古老的天文学、数学著作，公元/r/n3/r/n世纪初中国数学家赵爽创制了“勾股圆方图”/r/n(/r/n如图/r/n)/r/n，用以证明其中记载的勾股定理/r/n./r/n现提供/r/n4/r/n种不同颜色给右图中/r/n5/r/n个区域涂色，规定每个区域只涂一种颜色，相邻区域颜色不同，则不同涂色的方法种数为/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根据题意，分/r/n2/r/n步进行分析：/r/n

①对于区域/r/nABE/r/n，三个区域两两相邻，有/r/nA/r/n4/r/n3/r/n=24/r/n种涂色的方法，/r/n

②对于区域/r/nCD/r/n，若/r/nC/r/n区域与/r/nA/r/n颜色相同，/r/nD/r/n区域有/r/n2/r/n种选法，/r/n

若/r/nC/r/n区域与/r/nA/r/n颜色不同，则/r/nC/r/n区域有/r/n1/r/n种选法，/r/nD/r/n区域也只有/r/n1/r/n种选法，/r/n

则区域/r/nCD/r/n有/r/n2+1=3/r/n种涂色的方法，/r/n

则有/r/n24×3=72/r/n种涂色的方法，/r/n

故选：/r/nC./r/n

/r/n根据题意，分/r/n2/r/n步依次分析区域/r/nABE/r/n和区域/r/nCD/r/n的涂色方法数目，由分步计数原理计算可得答案。/r/n

本题考查排列组合的应用，涉及分步计数原理的应用，属于基础题。/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间，按照宿州市疫情防控应急指挥部的要求，市教育体育局对各市直学校下发了有关疫情防控通知．某学校按市局通知要求，制定了错峰放学，错峰吃饭的具体防疫措施．高三年级一层楼有/r/nA/r/n、/r/nB/r/n、/r/nC/r/n、/r/nD/r/n、/r/nE/r/n、/r/nF/r/n六个班排队吃饭，/r/nA/r/n班必须排在第一位，且/r/nD/r/nA/r/n．/r/n20/r/n种/r/n /r/nB/r/n．/r/n56/r/n种/r/n /r/nC/r/n．/r/n72/r/n种/r/n /r/nD/r/n．/r/n40/r/n种/r/nC/r/n．/r/n根据题意，/r/nA/r/n班必须排在第一位，剩下/r/n5/r/n个班级安排在后面的/r/n5/r/n个位置即可，/r/n分/r/n2/r/n步进行分析：/r/n①/r/n将/r/nBCF/r/n三个班级全排列，排好后有/r/n4/r/n个空位，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n种排法，/r/n②/r/n在/r/n4/r/n个空位中选出/r/n2/r/n个，安排/r/nD/r/n班、/r/nE/r/n班，有/r/nA/r/n42/r/n＝/r/n12/r/n种排法，/r/n则有/r/n6/r/n×/r/n12/r/n＝/r/n72/r/n种不同的方案/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的展开式中/r/nx/r/n3/r/ny/r/n3/r/n项的系数为（）/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故展开式中/r/nx/r/n3/r/ny/r/n3/r/n项的系数为/r/n•/r/n2/r/n2/r/n+2/r/n•/r/n2/r/n3/r/n＝/r/n200/r/n．/r/n13.(2021•安徽马鞍山三模•理T/r/n6/r/n．)/r/n的常数项为/r/n25/r/n，则实数/r/na/r/n的值为（）/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的展开式的通项公式为/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的常数项为（﹣/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•河北邯郸二模•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的展开式中/r/nx/r/n3/r/n项的系数为（）/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因为根据（/r/n1+/r/nx/r/n）/r/n6/r/n展开式的通项/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的展开式中/r/nx/r/n3/r/n项为/r/n＝﹣/r/n9/r/nx/r/n3/r/n，/r/n所以含/r/nx/r/n3/r/n项的系数﹣/r/n9/r/n．/r/n15.(2021•河北秦皇岛二模•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，则下列选项正确的是（）/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．展开式中系数最大的为/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则/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错误，/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则（/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正确，/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错误，/r/n展开式中偶数项系数为负值，奇数项系数为正值，/r/n则系数最大的在/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展开式的通项公式/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则/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则系数最大的为/r/na/r/n2/r/n．/r/n16.(2021•辽宁朝阳二模•T6．)/r/n今年我国中医药选出的“三药三方”对治疗新冠肺炎均有显著效果，功不可没．“三药”分别为金花清感颗粒、连花清瘟胶囊、血必清注射液；“三方”分别为清肺排毒汤、化湿败毒方、宜肺败毒方，若某医生从“三药三方”中随机选出/r/n2/r/n种，则恰好选出/r/n1/r/n药/r/n1/r/n方的方法种数为（）/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根据题意，某医生从“三药三方”中随机选出/r/n2/r/n种，恰好选出/r/n1/r/n药/r/n1/r/n方，/r/n则/r/n1/r/n药的取法有/r/n3/r/n种，/r/n1/r/n方的取法也有/r/n3/r/n种，/r/n则恰好选出/r/n1/r/n药/r/n1/r/n方的方法种数为/r/n3/r/n×/r/n3/r/n＝/r/n9/r/n．/r/n17.(2021•山东潍坊二模•T/r/n4/r/n．)/r/n2021/r/n年是中国共产党百年华诞．某学校社团将举办庆祝中国共产党成立/r/n100/r/n周年革命歌曲展演．现从《歌唱祖国》《英雄赞歌》《唱支山歌给党听》《毛主席派人来》/r/n4/r/n首独唱歌曲和《没有共产党就没有新中国》《我和我的祖国》/r/n2/r/n首合唱歌曲中共选出/r/n4/r/n首歌曲安排演出，要求最后一首歌曲必须是合唱，则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根据题意，在/r/n2/r/n首合唱歌曲中任选/r/n1/r/n首，安排在最后，有/r/n2/r/n种安排方法，/r/n在其他/r/n5/r/n首歌曲中任选/r/n3/r/n首，作为前/r/n3/r/n首歌曲，有/r/nA/r/n5/r/n3/r/n＝/r/n60/r/n种安排方法，/r/n则有/r/n2/r/n×/r/n60/r/n＝/r/n120/r/n种不同的安排方法/r/n．/r/n1/r/n8/r/n.(2021/r/n•浙江丽水湖州衢州二模•/r/nT8/r/n．/r/n)/r/n某/r/n市抽调/r/n5/r/n位医生分赴/r/n4/r/n所医院支援抗疫，要求每位医生只能去一所医院，每所医院至少安排一位医生．由于工作需要，甲、乙两位医生必须安排在不同的医院，则不同的安排种数是（）/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根据题意，分/r/n2/r/n步进行分析：/r/n①/r/n将/r/n5/r/n位医生分为/r/n4/r/n组，要求甲乙不在同一组，有/r/nC/r/n5/r/n2/r/n﹣/r/n1/r/n＝/r/n9/r/n种分组方法，/r/n②/r/n将分好的/r/n4/r/n组安排到/r/n4/r/n所医院支援抗疫，有/r/nA/r/n4/r/n4/r/n＝/r/n24/r/n种安排方法，/r/n则有/r/n9/r/n×/r/n24/r/n＝/r/n216/r/n种安排种数．/r/n二、填空题部分/r/n1/r/n9/r/n.(2021/r/n•山东潍坊二模•/r/nT13/r/n．/r/n)/r/n设/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，则/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•辽宁朝阳二模•T13．)在/r/n（/r/nx/r/n﹣/r/n2/r/ny/r/n+/r/nz/r/n）/r/n7/r/n的展开式中，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）的项的系数之和是/r/n/r/n．/r/n﹣/r/n21/r/n．/r/n因为（/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所以展开式中含/r/nz/r/n2/r/n的项为/r/nC/r/n，/r/n令/r/nx/r/n＝/r/ny/r/n＝/r/nz/r/n＝/r/n1/r/n，则所求系数之和为/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•广东潮州二模•/r/nT13/r/n．/r/n)/r/n（/r/nx/r/n﹣/r/n）/r/n6/r/n的展开式的常数项是/r/n/r/n．用数字作答）．/r/n﹣/r/n20/r/n．/r/n（/r/nx/r/n﹣/r/n）/r/n6/r/n的展开式的通项公式为/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，可得展开式的常数项为/r/n•（﹣/r/n1/r/n）＝﹣/r/n20/r/n．/r/n22/r/n.(2021/r/n•广东潮州二模•/r/nT15/r/n．/r/n)/r/n根/r/n据中央关于精准脱贫的要求，我市农业经济部门随机派遣/r/n4/r/n位专家对/r/n3/r/n个县区进行调研，每个县区至少派/r/n1/r/n位专家，则专家派遣的方法的种数为/r/n/r/n．/r/n36/r/n．/r/n根据题意，分/r/n2/r/n步进行分析：/r/n①/r/n将/r/n4/r/n位专家分为/r/n3/r/n组，有/r/nC/r/n4/r/n2/r/n＝/r/n6/r/n种分组方法，/r/n②/r/n将分好的三组全排列，分到/r/n3/r/n个县区进行调研，有/r/nA/r/n3/r/n3/r/n＝/r/n6/r/n种情况，/r/n则有/r/n6/r/n×/r/n6/r/n＝/r/n36/r/n种派遣方法/r/n．/r/n23.(2021•天津南开二模•T/r/n11/r/n．)/r/n的二项展开式中，/r/nx/r/n3/r/n的系数等于/r/n/r/n．/r/n15/r/n．/r/n的二项展开式的通项公式为/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的系数等于（﹣/r/n5/r/n）/r/n2/r/n＝/r/n15/r/n．/r/n24.(2021•浙江卷•T13)/r/n已知多项式/r/n，则/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故答案为./r/n25.(2021•江苏盐城三模•T/r/n13/r/n)/r/n文旅部在/r/n2021/r/n年围绕/r/n“/r/n重温红色历史、传承奋斗精神/r/n”“/r/n走进大国重器、感受中国力量/r/n”“/r/n体验美丽乡村、助力乡村振兴/r/n”/r/n三个主题，遴选推出/r/n“/r/n建党百年红色旅游百条精品线路/r/n”/r/n．这些精品线路中包含上海/r/n—/r/n大会址、嘉兴南湖、井冈山、延安、西柏坡等/r/n5/r/n个传统红色旅游景区，还有港珠澳大桥、北京大兴国际机场、/r/n“/r/n中国天眼/r/n”/r/n、/r/n“/r/n两弹一星/r/n”/r/n纪念馆、湖南十八洞村、浙江余村、贵州华茂村等/r/n7/r/n个展现改革开放和新时代发展成就、展示科技强国和脱贫攻坚成果的景区．为安排旅游路线，从上述/r/n12/r/n个景区中选/r/n3/r/n个景区，则至少含有/r/n1/r/n个传统红色旅游景区的选法有种．/r/n185/r/n．/r/n【考点】排列组合/r/n由题意，可用间接法，总体情况为从/r/n12/r/n个景区中选/r/n3/r/n个景区/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，从/r/n7/r/n个非传统红色旅游景区中选/r/n3/r/n个景区/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，则至少含有/r/n1/r/n个传统红色旅游景区的选法有/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•河南郑州三模•理T/r/n14/r/n)/r/n展开式中的/r/na/r/n与/r/nb/r/n指数相同的项的表达式为/r/n/r/n．/r/n﹣/r/n84/r/na/r/n．/r/n展开式的通项公式为/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所以展开式中的/r/na/r/n与/r/nb/r/n指数相同的项的表达式为/r/nC/r/n＝﹣/r/n84/r/na/r/n．/r/n27.(2021•四川内江三模•理T/r/n14/r/n．)/r/n二项式（/r/n2/r/nx/r/n2/r/n﹣/r/n）/r/n6/r/n的展开式中的常数项是/r/n/r/n．（用数字作答）/r/n60/r/n．/r/n（/r/n2/r/nx/r/n2/r/n﹣/r/n）/r/n6/r/n的展开式的通项公式为/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展开式中的常数项为/r/n﹣/r/n19/r/n/r/n．/r/n﹣/r/n19/r/n．/r/n依题意，（/r/n1/r/n﹣/r/nx/r/n）/r/n6/r/n展开式的通项是/r/n，/r/n当/r/nr/r/n＝/r/n0/r/n时，/r/n；/r/n当/r/nr/r/n＝/r/n3/r/n时，/r/n，/r/n∴/r/n展开式的常数项是/r/n．/r/n29.(2021•四川泸州三模•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展开式中/r/nx/r/n3/r/n项的系数为/r/n/r/n．/r/n﹣/r/n5/r/n．/r/n由题意可得展开式中含/r/nx/r/n3/r/n项为/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•江苏常数三模•T/r/n13/r/n．)/r/n的展开式中常数项为/r/n/r/n．/r/n﹣/r/