How many 7card hands dealt are there with three pairs each o

How many 7-card hands dealt are there with three pairs (each of a different kind, plus a seventh card of a different kind) ?

Solution

I begin by addressing the 3 pairs of different kinds. There are a total of 13 denominations of which I shall select 3 denominations. Each denomination has 4 cards. Out of my 3 selected denominations I shall pick 2 cards for a pair.

There are C(13,3) * C (4,2)^3 ways to pick the six cards

The seventh card is also suppossed to be a different kind, so having already chosen three kinds or denominations, wouldn\'t that eliminate 12 cards leaving me with 40 choices for the last card

so,

40*C(13,3) * C (4,2)^3