The number of combination of n distinct objects taken r at a time be x is given by

ⁿC_r = n!/{(n - r)! * r!}

Let the number of combination of n distinct objects taken r at a time be x.

Now consider one of these n ways. There are e objects in this selection which can be arranged in r! ways.

So, each of the x combinations gives rise to r! permutations. So, x combinations will give rise to x*(r!).

Consequently, the number of permutations of n things, taken r at a time is x*(r!) and it is equal to ⁿP_r 

So,  x*(r!) = ⁿP_r

=> x*(r!) = n!/(n - r)!

=> x = n!/{(n - r)! * r!} 

=> ⁿC_r = n!/{(n - r)! * r!}