Solution:

The condition says $\binom{n}{6} = 6\binom{n}{3}.$ Dividing the binomial coefficients, $$\frac{\binom{n}{6}}{\binom{n}{3}} = \frac{(n-3)(n-4)(n-5)}{6 \cdot 5 \cdot 4} = 6,$$ so $(n-3)(n-4)(n-5) = 720 = 10 \cdot 9 \cdot 8.$

Since the product $(n-3)(n-4)(n-5)$ is increasing in $n,$ the only solution is $n - 3 = 10,$ that is, $n = 13.$