If $^{n} P_{n} = 5040$ , find ' $n$ '

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Solution

$^{n} P_{n} = 5040$
$\Rightarrow^{n} P_{n} = n! = 5040$
Factors of $5040 \to 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7$
This can be written as $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Which is $7!$
$\Rightarrow^{n} P_{n} = 7! = n!$
$∴ n = 7$
Hence, the answer is $7.$

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