HMMT 二月 2006 · COMB 赛 · 第 5 题

英文解析

5. Fifteen freshmen are sitting in a circle around a table, but the course assistant (whoremains standing) has made only six copies of today’s handout. No freshman shouldget more than one handout, and any freshman who does not get one should be able toread a neighbor’s. If the freshmen are distinguishable but the handouts are not, howmany ways are there to distribute the six handouts subject to the above conditions?
   Answer: 125
   Solution: Suppose that you are one of the freshmen; then there’s a 6 / 15 chance thatyou’ll get one of the handouts. We may ask, given that you do get a handout, howmany ways are there to distribute the rest? We need only multiply the answer to thatquestion by 15 / 6 to answer the original question.
   Going clockwise around the table from you, one might write down the sizes of the gapsbetween people with handouts. There are six such gaps, each of size 0–2, and the sumof their sizes must be 15 − 6 = 11. So the gap sizes are either 1, 1, 1, 2, 2, 2 in some
   6!
   order, or 0, 1, 2, 2, 2, 2 in some order. In the former case, = 20 orders are possible;
   3!3!
   6!
   in the latter, = 30 are. Altogether, then, there are 20 + 30 = 50 possibilities.
   1!1!4!
   Multiplying this by 15 / 6, or 5 / 2, gives 125.