HMMT 二月 2003 · GEN1 赛 · 第 6 题

英文解析

6. In how many ways can 3 bottles of ketchup and 7 bottles of mustard be arranged in arow so that no bottle of ketchup is immediately between two bottles of mustard? (Thebottles of ketchup are mutually indistinguishable, as are the bottles of mustard.)
   Solution: 22
   Consider the blocks of consecutive bottles of ketchup in such an arrangement. A blockof just one bottle must occur at the beginning or the end of the row, or else it would be between two bottles of mustard. However, a block of two or three bottles can occuranywhere. We cannot have three blocks of one bottle each, since there are only twopossible locations for such blocks. Thus, we either have a block of one bottle and ablock of two, or one block of all three bottles. In the first case, if the single bottleoccurs at the beginning of the row, then anywhere from 1 to 7 bottles of mustard mayintervene before the block of 2 ketchup bottles, giving 7 possible arrangements. Welikewise have 7 arrangements if the single bottle occurs at the end of the row. Finally,
   if there is just one block of three bottles, anywhere from 0 to 7 mustard bottles mayprecede it, giving 8 possible arrangements. So, altogether, we have 7 + 7 + 8 = 22
   configurations.
   3 2