Five choose six equals zero

Algebra Level 4

Let $$h(x)$$ denote a monic $$8^\text{th}$$-degree polynomial such that $$h(m) = \binom m6$$ for $$m = 6,7,8,\ldots,13$$.

Find $$h(5)$$.

Notation: $$\binom mn$$ denotes the binomial coefficient $$\binom mn = \frac{m!}{n!(m-n)!}$$ for non-negative integers $$m\geq n$$.

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