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Question

If m and n are positive integers greater than or equal to 2, m > n, then (mn)! is divisible by


A

(m!)n

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B

(n!)m

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C

(m+n)!

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D

(mn)!

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Solution

The correct options are
A

(m!)n


B

(n!)m


C

(m+n)!


D

(mn)!


(mn)!(m!)n is the number of ways of distributing mn distinct objects in n persons equally. Hence (mn)!(m!)n is an integer

(m!)n|(mn)!. Similarly (n!)m|(mn)!

Further m+n<2mmn(m+n)!|(mn)! and mn<m<mn

(mn)!|(mn)!


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