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Let $$P(x) = \sum_{r=0}^{7} a_r x^r$$, where $$a_7 = 1$$ and $$P(k) = k$$ for $$k = 1, 2, 3, \ldots, 7$$ then $$P(8) =$$
$$P(x) - x$$ is a degree 7 polynomial with leading coefficient 1 that vanishes at $$x = 1, 2, \ldots, 7$$, so $$P(x) - x = (x - 1)(x - 2) \cdots (x - 7)$$. Then $$P(8) = 7! + 8 = 5040 + 8 = 5048$$.
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