First slide

Permutations

Question

If $0 < r < s \leq n$ and $^{n} P_{r} =^{n} P_{s}$ then value of $r + s$ is

Moderate

A

$2 n - 2$
B

$2 n - 1$
C

2
D

1

Solution

$^{n} P_{r} =^{n} P_{s} \Rightarrow \frac{n!}{(n - r)!} = \frac{n!}{(n - s)!}$
$\Rightarrow (n - r)! = (n - s)!$
As $r < s, n - r > n - s$ But the only two different factorials which are equal are $0!$ and $1!$ Thus $n - r = 1$ and $n - s = 0$
$\begin{aligned} \Rightarrow r = n - 1 and s = n . \\ \Rightarrow r + s = 2 n - 1 . \end{aligned}$

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