公务员值得学习排列与组合公式课程.doc

排列数公式 ==.(,∈N*,且).注:规定. 排列恒等式 (1); (2); (3); (4); (5). (6) . 组合数公式 ===(∈N*,,且).组合数的两个性质(1)= ; (2) +=. 注:规定. 组合恒等式(1); (2); (3); (4)=; (5). (6). (7). (8). (9). (10). 排列数与组合数的关系 . 单条件排列 以下各条的大前提是从个元素中取个元素的排列. (1)“在位”与“不在位” ①某(特)元必在某位有种;②某(特)元不在某位有(补集思想)(着眼位置)(着眼元素)种. (2)紧贴与插空(即相邻与不相邻) ①定位紧贴:个元在固定位的排列有种. ②浮动紧贴:个元素的全排列把k个元排在一起的排法有种.注:此类问题常用捆绑法; ③插空:两组元素分别有k?h个(),把它们合在一起来作全排列,k个的一组互不能挨近的所有排列数有种. (3)两组元素各相同的插空 个大球个小球排成一列,小球必分开,问有多少种排法? 当时,无解;当时,有种排法. (4)两组相同元素的排列:两组元素有m个和n个,各组元素分别相同的排列数为. 158.分配问题 (1)(平均分组有归属问题)将相异的?个物件等分给个人,各得件,其分配方法数共有. (2)(平均分组无归属问题)将相异的·个物体等分为无记号或无顺序的堆,其分配方法数共有 . (3)(非平均分组有归属问题)将相异的个物体分给个人,物件必须被分完,分别得到,,…,件,且,,…,这个数彼此不相等,则其分配方法数共有. (4)(非完全平均分组有归属问题)将相异的个物体分给个人,物件必须被分完,分别得到,,…,件,且,,…,这个数中分别有a?b?c?…个相等,则其分配方法数有 . (5)(非平均分组无归属问题)将相异的个物体分为任意的,,…,件无记号的堆,且,,…,这个数彼此不相等,则其分配方法数有. (6)(非完全平均分组无归属问题)将相异的个物体分为任意的,,…,件无记号的堆,且,,…,这个数中分别有a?b?c?…个相等,则其分配方法数有. (7)(限定分组有归属问题)将相异的()个物体分给甲?乙?丙,……等个人,物体必须被分完,如果指定甲得件,乙得件,丙得件,…时,则无论,,…,等个数是否全相异或不全相异其分配方法数恒有 . 159.“错位问题”及其推广 贝努利装错笺问题:信封信与个信封全部错位的组合数为 . 推广: 个元素与个位置,其中至少有个元素错位的不同组合总数为 Acknowledgments The authors would like to thank Johns Hopkins University for the TC-1 cells. This work was supported by a National Health Research Institutes intramural grant (IV-103-PP-22) and grants from the National Science Council, which were awarded to Y.C. Song (NSC 99-2321-B-400-004-MY3) and S.J. Liu (NSC 103-2321-B-400-008). Author Contributions Y.C.S. and S.J.L. designed the studies. Y.C.S. performed the research and analyzed the data. Y.C.S. and S.J.L. wrote the manuscript. Additional Information C57BL/6 mice were immunized subcutaneously (s.c.) once with 1 μ g of peptide mixed with or without 10 μ g of CpG adjuvant. After one week, splenocytes were harvested, and the response of IFN-γ -secreting cells was determined by ELISPOT after 48 h of peptide stimulation. Briefly, 2 × 105 splenocytes were incubated with 1 μ g/ml irrelevant peptide or RAH peptide in an anti-IFN-γ -coated polyvinylidene fluoride (PVDF) plate for 48 h. Af