HMMT 十一月 2018 · 冲刺赛 · 第 5 题

HMMT November 2018 — Guts Round — Problem 5

[ 6 ] Jerry has ten distinguishable coins, each of which currently has heads facing up. He chooses one coin and flips it over, so it now has tails facing up. Then he picks another coin (possibly the same one as before) and flips it over. How many configurations of heads and tails are possible after these two flips? ◦ ◦ ◦ ◦ ◦ ◦

解析

[ 6 ] Jerry has ten distinguishable coins, each of which currently has heads facing up. He chooses one coin and flips it over, so it now has tails facing up. Then he picks another coin (possibly the same one as before) and flips it over. How many configurations of heads and tails are possible after these two flips? Proposed by: Andrew Gu Answer: 46 We have two cases: Case 1: Jerry picks the same coin twice. Then, the first time he flips the coin, it becomes tails, and then the second time, it becomes heads again, giving us the original state of all heads. Case 2: Jerry picks two di ↵ erent coins. In this case, there are two coins with tails face up, and the 10 10 · 9 rest are heads face up. There are = = 45 ways to pick which two coins have tails. 2 2 Adding up the possibilities from both cases, we have a total of 1 + 45 = 46 possible configurations.