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组合研究的是无次序的选取问题,所谓一个组合也就是从n个不同元素中不计顺序地选取m个构成原来集合的一个子集。例1 有5双共10只尺码不同的手套(左、右手有区别),从这10只手套中取出4只。 (1) 恰有2双的取法有多少种? (2) 恰有2只成一双的取法有多少种? 分析 (1)从5双手套中取2双,这2双手套间彼此不计较顺序,故有C_5~2=10种不同取法。 (2)先从5双手套中取一双有C_5~1种取法,再从剩下的4双即8只手套中取2只,要求不属同一双,可分两步:先从8只中取1只,除去与它同一双的另一只,再从其它6只中取一只,有C_6~1·C_6~1种取法,不过这里面有重复。为说明问题,不妨设有两双手套:A_1、B_1与A_2,B_2,其中先取出A_1再取出A_2与先取出A_2再取出A_1
The combinatorial research is the problem of unordered selection. A combination is to select m out of n different elements without order to form a subset of the original set. Example 1 There are 5 pairs of 10 different size gloves (left and right hand are different). Take out 4 of the 10 gloves. (1) How many kinds of two pairs are there? (2) How many kinds of two are in a single pair? Analysis (1) Take two pairs of five pairs of gloves. The two pairs of gloves do not count one another. Therefore, there are C_5~2=10 different methods. (2) Take a pair of C_5~1 methods from the 5 pairs of gloves, and then take 2 from the remaining 4 pairs, ie 8 gloves. The requirements are not in the same pair. It can be divided into two steps: first from 8 Take one, remove the other one with the same pair, and then take one from the other six. There are C_6~1•C_6~1 methods, but there are repetitions. To illustrate the problem, two pairs of gloves may be provided: A_1, B_1, and A_2, B_2, where A_1 is first taken out, then A_2 is taken out, A_2 is taken out first, and then A_1 is taken out.