排列组合问题怎么判断要不要考虑顺序?比如35题不用考虑学者顺序,而36题第2小题为什么又要考虑三个

2025-03-21 10:46:54
推荐回答(1个)
回答1:

其实两道题都考虑了顺序的问题,唯一不同的就是35题中10本书是有3个3本书,也就是说涉及到了一个平分的问题,这里我们先讨论一下平分的问题:
先举个简单的例子,序号为1-4的4个大小相同小球,先准备均分为两堆,有几种分法?
这里直接用列举法就可得知答案,12,34;13,24;14,23;一共就三堆,那么再讨论一下一般解法,4个球先选2个一堆4C2,在2个球选2个一堆2C2,那么一共是4C2*2C2=6种情况,这里就有一个问题,用4C2*2C2解出来的值偏大,这里再解释一个,其实4C2*2C2里面有大量的重复情况,比如4C2里选出了13,那么对应的就是24,构成一种分发,这时候4C2里也能选出24,那么对应的就是13,在均分两堆不考虑顺序的情况下,13,24;24,13是同一种情况,所以是有重复的,因为4C2*2C2里面已经考虑了排列的情况,所以要平分的话需要再除以2堆的全排列情况也就是2A2
同理在9个球均分3堆,则是9C3*6C3*3C3/3A3
那么其实35题就很好理解了,10本书的分法,先10本里选一本10C1,剩下9本均分3堆即9C3*6C3*3C3/3A3,所以情况是10C1*9C3*6C3*3C3/3A3,4个人再全排列再乘以4A4,所以得到的结果就是4C1*10C1*9C3*6C3*3C3,4C1也可以理解成4个人里先选一个是给1本书的,然后3个人分三堆书10C1*9C3*6C3*3C3种情况,所以答案就是这样,看起来好像没有涉及到排列。
其实你一开始的想法要考虑全排列是正确的,只是对平分问题理解还不够到位,错以为没有排列,不要因为这个题造成以后你对解题考虑排列或是组合的迷茫,这点很重要。

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