Pre-muted

Number Theory Level 2

You are given that n ! ( n − x ) ! = n \dfrac{n!}{(n-x)!} = n

Find x x .

Bonus : Can you also find one exception for the above formula?

Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .


The answer is 1.

Viki Zeta by Viki Zeta , 19, India

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2 solutions

Viki Zeta
Aug 11, 2016

Given that, n ! ( n − x ) ! = n ⟹ n ! n = ( n − x ) ! ⟹ n ( n − 1 ) ! n = ( n − x ) ! ⟹ ( n − 1 ) ! 1 = ( n − x ) ! ⟹ ( n − 1 ) ! = ( n − x ) ! ⟹ ( n − 1 ) = ( n − x ) , Same numbers premute same result ⟹ − 1 = − x ⟹ x = 1 \text{Given that, }\\ \dfrac{n!}{(n-x)!} = n \\ \implies \dfrac{n!}{n} = (n-x)! \\ \implies \dfrac{n(n-1)!}{n} = (n-x)! \\ \implies \dfrac{(n-1)!}{1} = (n-x)! \\ \implies (n-1)! = (n-x)! \\ \implies (n-1) = (n-x) \text{, Same numbers premute same result} \\ \implies -1 = -x \\ \implies x = 1

Therefore, x = 1 x=1 . But, if you take n = 0 n=0 , this won't apply. On premutating 0 ! 0! , we get answer as 1 1 and not 0 0

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0! is one, but it is still not right because (-1)! is not possible.

Lohith Tummala - 4 years, 9 months ago

What about the case when x=0 and n=1?

John Jo - 4 years, 8 months ago

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I've already made a note in the solution

Viki Zeta - 4 years, 8 months ago
Edwin Gray
Jul 16, 2018

Dinide both sides by n, resulting in (n -1)!/(n - x)! =1. So x =1. Ed Gray

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