From a deck of 52 cards, a 5 card hand is dealt.How many distinct hands can be formed if there are atleast 2 queens?

The total possible cases would be a 5 card hand with no restrictions : $52C5$ 5

The unwanted cases are:

no queens(out of 48 non-queens cards we get 5)  $48C5$

only 1 queen(out of 4 queens we get 1,and out of 48 non-queens we get 4)  $4C1*48C4$

Therefore, $52C5-(48C5+4C1*48C4) = 108336$