First slide

Permutations

Question

The product of n consecutive natural number is always divisible by

Moderate

A

$^{n} P_{n}$
B

$^{2 n} C_{n}$
C

$^{2 n} P_{n}$
D

$^{n + 1} P_{n}$

Solution

Let $n$ consecutive natural numbers $b$ $m + 1, m + 2 \dots, m + n$ where $m \geq 0$
$\begin{array}{l} P = (m + 1) (m + 2) \dots (m + n) \\ = (\frac{m! (m + 1) \dots (m + n)}{m! n!}) n! \\ = (^{m + n} C_{n}) (^{n} P_{n}) \end{array}$
$∴ P$ is divisible by $^{n} P_{n}$

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