HMMT February

Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:
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?

Solution

Solution:
For $0 \leq k \leq 6$ , to obtain a score that is $k \pmod{6}$ , 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 $7-k$ . So the total number of possible scores is $\sum_{k=0}^{6}(7-k)=28$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.