HMMT 十一月 2012 · 冲刺赛 · 第 14 题
HMMT November 2012 — Guts Round — Problem 14
题目详情
- [ 9 ] Mark would like to place the numbers 1 to 7 on a circle such that the sequences along both arcs going from 1 to 7 are increasing. For example, one arc could be 1, 2, 4, 7 and the other could be 1, 3, 5, 6, 7. In how many distinct ways can Mark place the numbers? Two arrangements are distinct if and only if one cannot be rotated to match the other.
解析
- [ 9 ] Answer: 32 First, we can fix the position of the 1. Then, by the condition that the numbers are increasing along each arc from 1, we know that the 2 must be adjacent to the 1; so we have two options for its placement. Similarly, we have two options for placing each of 3,4,5,6 in that order. Finally, the 5 7 must go in the remaining space, for a total of 2 = 32 orderings.