State whether the follwing statement is true or false. $(n + 1)^{n} P_{r} = (n - r + 1)^{(n + 1)} P_{r}$ .

True

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False

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Solution

The correct option is A True
${(n + 1)}^{n} P_{r} = {(n - r + 1)}^{(n + 1)} P_{r}$
LHS $= {(n + 1)}^{n} P_{r} = (n + 1) \frac{n!}{(n - r)!} = \frac{(n + 1)!}{(n - r)!}$
RHS $= (n - r + 1) \frac{(n + 1)!}{(n + 1 - r)!}$
LHS $= \frac{(n + 1)!}{(n - r)!}$
RHS $= \frac{(n + 1)!}{(n - r + 1) (n - r)!} (n - r + 1) = \frac{(n + 1)!}{(n - r)!}$
LHS=RHS
$∴$ True

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