HMMT 十一月 2012 · 冲刺赛 · 第 14 题
HMMT November 2012 — Guts Round — Problem 14
题目详情
英文原题
- [ 9 ] Mark would like to place the numbers 1 to 7 on a circle such that the sequences along both arcsgoing 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 ifand only if one cannot be rotated to match the other.
解析
英文解析
- [ 9 ]
Answer: 32 First, we can fix the position of the 1. Then, by the condition that the numbers areincreasing along each arc from 1, we know that the 2 must be adjacent to the 1; so we have two optionsfor its placement. Similarly, we have two options for placing each of 3,4,5,6 in that order. Finally, the
7 must go in the remaining space, for a total of 2 = 32 orderings.5