2019 Mock AMC 10B Problems/Problem 16

Solution by excruciating: For each of $a,b,c,d,e,f,g$ , they all need to be at least $1$ , so we have: $7 +a+b+c+d+e+f+g\leq n,$ or $a+b+c+d+e+f+g \leq n-7$ without the " $a,b,c,d,e,f,g$ need to be $>0$ restriction." So now it's stars and bars:

${n-7+7-1 \choose 7-1},$ which simplifies to ${n-1 \choose 6},$ which we can write as $\frac{(n-1)!}{6! \cdot (n-7)!} = 3003$ .

Expanding, we get $(n-1)(n-2)(n-3)(n-4)(n-5)(n-6) = 3003 \cdot 6! = 2^4 \cdot 3^3 \cdot 5 \cdot 7 \cdot 11\ cdot 13$ .

Because we only have $6$ terms, $(n-1)$ is to be around $13,$ the highest divisor. Thus, we must have $5\cdot2 = 10$ as a divisor too. $2^2 \cdot 3 = 12$ is one too. We have left is $2, 3^2,$ and $7$ , to make $2$ terms, so we have $3^2 = 9$ , and $7\cdot2 = 14$ . Thus we have: $(n-1)(n-2)(n-3)(n-4)(n-5)(n-6) = 14\cdot13\cdot12\cdot11\cdot10\cdot9.$

Thus, $n = 14 + 1 = \boxed{(E) 15}.$

Note: $g$ does not need to be $\geq 1$ because the inequality $a+b+c+d+e+f\leq n$ is not strict. So this solution is incorrect.

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2019 Mock AMC 10B Problems/Problem 16