HMMT 二月 2011 · CALCCOMB 赛 · 第 1 题

HMMT February 2011 — CALCCOMB Round — Problem 1

A classroom has 30 students and 30 desks arranged in 5 rows of 6. If the class has 15 boys and 15 girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

解析

A classroom has 30 students and 30 desks arranged in 5 rows of 6. If the class has 15 boys and 15 girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl? 2 Answer: 2 · 15! If we color the desks of the class in a checkerboard pattern, we notice that all of one gender must go in the squares colored black, and the other gender must go in the squares colored white. There are 2 ways to pick which gender goes in which color, 15! ways to put the boys into desks 2 and 15! ways to put the girls into desks. So the number of ways is 2 · 15! . (There is a little ambiguity in the problem statement as to whether the 15 boys and the 15 girls are distinguishable or not. If they are not distinguishable, the answer is clearly 2. Given the number of contestants who submitted the answer 2, the graders judged that there was enough ambiguity to justify 2 accepting 2 as a correct answer. So both 2 and 2 · 15! were accepted as correct answers.)