why in a combination the denominator is multiplied by r?

The formula for combinations is a modification of the formula for permutations.

The permutation formula:  _nP_r  =  n! / (n - r)!

With permutations, order is important; with combinations, order is not important.

With permutations, these sets of three letters are different:  ABC, ACB, BAC, BCA, CAB, CBA.

With combinations, these sets are not different, because their order is not important.

So these 6 sets in a permutation create only 1 set in a combination.

Thus, you have to divide the answer in the permutation by the number of ways that the selected letters can be arranged.

The number of ways that r elements can be arranged is r!.

So divide the permutation answer by r! to get the combinations answer:  _nC_r  =  n! / [ (n - r)! · r! ]