Are they integers?

Probability Level 3

Let $S=\dfrac{(mn)!}{(m!)^{n}}$ , where $m$ and $n$ are positive integers. We know that $S$ is always an integer. Are $\dfrac{S}{m!}$ and $\dfrac{S}{n!}$ always integers, too?

Only

\dfrac{S}{m!}

is. Both of them are. Neither of them are. Only

\dfrac{S}{n!}

is.

1 solution

X X
Apr 21, 2018

$S$ is the number of the ways of putting $mn$ different balls into $n$ different boxes,and each box contain $m$ balls.But if the $n!$ boxes are identical,then the number of the ways of putting $mn$ different balls in them will be $\frac{S}{n!}$ .So, $\frac{S}{n!}$ must be an integer.About $\frac{S}{m!}$ ,put $m=3,n=2$ ,and $\frac{S}{m!}=\frac{10}{3}$ will not be an integer.

Are they integers?

1 solution

0 pending reports