HMMT 二月 2016 · 冲刺赛 · 第 7 题

HMMT February 2016 — Guts Round — Problem 7

[ 6 ] A contest has six problems worth seven points each. On any given problem, a contestant can score either 0, 1, or 7 points. How many possible total scores can a contestant achieve over all six problems?

解析

[ 6 ] A contest has six problems worth seven points each. On any given problem, a contestant can score either 0, 1, or 7 points. How many possible total scores can a contestant achieve over all six problems? Proposed by: Evan Chen Answer: 28 For 0 ≤ k ≤ 6, to obtain a score that is k (mod 6) exactly k problems must get a score of 1. The remaining 6 − k problems can generate any multiple of 7 from 0 to 7(6 − k ), of which there are 7 − k . 6 ∑ So the total number of possible scores is (7 − k ) = 28. k =0