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对于抽象的排列组合问题,可以变化思维的角度,将它们转化为容易分析的等价形式. 1.利用对称性 例1 同室四人各写一张贺年卡,先集中起来,然后每人从中拿一张别人送出的贺年卡,则四张贺年卡不同的分配方式有() (A)6种(B)9种(C)11种(D)23种 解利用对称性,不妨让甲先拿,则甲有3种取法,假设甲取乙的贺年卡,然后让乙再取,则乙有3种取法,而丙,丁只能互拿对方的贺年卡.所以共有3·3=9种分配方式,选B. 2.利用可逆性 例2 马路上有编号为 1,2,3,…,10的十只路灯,为节约用电而又不影响照明,可以把其中的3只路灯熄掉,但不能同时熄掉相邻的两只或三只路灯.问满足条件的熄掉三只路灯的方法有_种. 解熄掉三只和熄掉三支后依照同样要
For the problem of abstract arrangement and combination, we can change the perspective of thinking and turn them into an equivalent form that can be easily analyzed. 1. Using Symmetry Example 1 Each roommate writes a New Year’s card and gathers them first, then each person takes it from For a new year’s greeting card sent by others, the four different greeting cards have different distribution methods () (A) 6 (B) 9 (C) 11 (D) 23 kinds of solutions using symmetry, may wish to let A first , there are three kinds of methods for A, assuming that A takes B’s New Year’s card, and then B gets it again, then B has 3 kinds of methods, while C and D can only get each other’s new year cards. So there are 3 3 = 9 types. Distribution method, choose B. 2. Use reversibility Example 2 There are ten street lamps numbered 1,2,3,...,10 on the street. In order to save electricity without affecting lighting, three of the road lamps can be turned off. But you can’t extinguish two or three street lights at the same time. Ask how to extinguish the three street lights that meet the conditions. There are _ ways to solve the conditions. Excite the three out and follow the same