The value of ^{n – 1}P_r + r ^{n – 1}P_r_{– 1} is __________.

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Solution

$^{n - 1} P_{r} + r^{n - 1} P_{r - 1} = \frac{(n - 1)!}{(n - 1 - r)!} + r \frac{(n - 1)!}{(n - 1 - r + 1)!} = \frac{(n - 1)!}{(n - r - 1)!} + r \frac{(n - 1)!}{(n - r)!} = (n - 1)! [\frac{1}{(n - r - 1)!} + \frac{r}{(n - r) (n - r - 1)!}] = (n - 1)! [\frac{n - r + r}{(n - r) (n - r - 1)!}] = \frac{(n - 1)! (n - r + r)}{(n - r)!} = \frac{(n - 1)! n}{(n - r)!} =^{n} P_{r}$

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