Five choose six equals zero

Algebra Level 4

Let h(x)h(x) denote a monic 8th8^\text{th} -degree polynomial such that h(m)=(m6)h(m) = \binom m6 for m=6,7,8,,13m = 6,7,8,\ldots,13.

Find h(5)h(5) .

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

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