# Bridge hand Combination/Permutation

A Bridge hand consists of 13 cards from a deck of 52 cards. In how many ways can a (bridge) hand consisting of 5 spades(♠), 4 hearts(♥), 4 diamonds(♦) and 0 clubs(♣) be selected?

There are $13 \choose 5$ ways of choosing $5$ spades.

There are $13 \choose 4$ ways of choosing $4$ hearts.

There are $13 \choose 4$ ways of choosing $4$ diamonds.

So there are $13 \choose 5$$\times$$13 \choose 4$$\times$$13 \choose 4$ ways of selecting a bridge hand with $5$ spades, $4$ hearts, $4$ diamonds, and $0$ clubs.

A slightly different answer is also possible, depending on whether the question is "how many ways can a (bridge) hand ... be selected", or whether it means how many possible different hands there are. The previous answer is how many possible different hands there are. But how many ways you can select those hands would be 13P5 * 13P4 * 13P4 (not 13C5 * 13C4 * 13C4). Edit 20160416: Thanks for the comment Thomas - silly mistake by me. It is indeed (13C5*13C4*13C4)*13C1 as you noted.

• This is in error, the number of ways of selecting a hand is $13!$ times the other hands, while your answer is $5!4!4!$ times the original answer. Jan 17, 2016 at 14:17