this is a very famous problem.

suppose n is a natural number.

prove that    is always an integer.

if the question is not visible click on view.

or i am typing it here too(n!)!/[(n!)^(n-1)!] is an integer.

suppose i have n no. of apples, n no. of oranges,n of bananas and so on such that i have (n-1)! no. of different kinds of fruits in which each kind is having n no. of fruits.

so,the total no. of fruits m having is n+n+n+n.......(upto (n-1)! terms) = n(n-1)! = n!

now if someone asks me  how many arrangements of all these fruits taken all at a time is possible, then i would do this

      { since total no. of fruits is n! and there are (n-1)! different kinds of fruit and each kind contains "n " no of fruits}

And we know  no. of  arrangements  always comes out to be an integer therefore its an integer.

awesome method!

i didn't use the fruits though.

rest is the same...