HMMT 二月 2000 · GEN 赛 · 第 24 题

HMMT February 2000 — GEN Round — Problem 24

P eter is randomly lling b o xes with andy . If he has 10 pie es of andy and 5 b o xes in a ro w lab eled A, B, C, D, and E, ho w man y w a ys an he distribute the andy so that no t w o adja en t b o xes are empt y?

解析

If there are no empt y b o xes, there are 126 w a ys of distributing the iden ti al andy . If there is one empt y b o x, there are 84 w a ys of distributing the andy and 5 w a ys of ho osing the empt y b o x, so there are 84 5 = 420 w a ys. If there are t w o empt y b o xes, there are 36 w a ys of distributing the andy and 6 om binations of assigning the empt y b o xes so that they are not adja en t, so there are 36 6 = 216 w a ys. If there are three empt y b o xes, there are 9 w a ys of distributing the andy , and only 1 w a y of arranging the empt y b o xes. Ha ving four or v e empt y b o xes is imp ossible, sin e some t w o w ould ha v e to b e adja en t. So, the total n um b er of w a ys is 126 + 420 + 216 + 9 = 771 . 1