HMMT 十一月 2025 · THM 赛 · 第 5 题

HMMT November 2025 — THM Round — Problem 5

Compute the number of ways to erase 26 letters from the string SUNSUNSUNSUNSUNSUNSUNSUNSUNSUN such that the remaining 4 letters spell SUNS in order.

解析

Compute the number of ways to erase 26 letters from the string SUNSUNSUNSUNSUNSUNSUNSUNSUNSUN such that the remaining 4 letters spell SUNS in order. © 2025 HMMT Proposed by: Derek Liu 12 Answer: = 495 4 Solution: The remaining letters divide the 26 erased letters into five (possibly empty) strings, with the first four having length congruent to 0 mod 3 and the last having length congruent to 2 mod 3. See the example below; remaining letters are boxed, and the five strings have lengths 3, 6, 0, 15, and 2. SUNSUNSUNSUNSUNSUNSUNSUNSUNSUN If we write the lengths of these five strings as as 3 a , 3 a , 3 a , 3 a , and 3 a + 2, then their sum is 0 1 2 3 4 equal to 26, the number of erased letters. Hence, 3( a + a + a + a + a ) + 2 = 26, so we seek the 0 1 2 3 4 number of nonnegative integer solutions to a + a + a + a + a = (26 − 2) / 3 = 8. By stars and bars, 0 1 2 3 4 8+4 12 the answer is = = 495 . 4 4 Remark. The alternative answer 495 · 26!, which can be obtained if a contestant took into account an ordering of deletion of the unused 26 digits, was also accepted.