Question

10. How many ways are there to choose 5 cards from 52 deck of cards to obtain a three of a kind?

Answer 1

Three of a kind means, there are three cards of the same rank and two cards of different ranks, such as 4,4,4,8,9 or 5,5,5,Q,K.
Now we want to find the number of ways to choose 5 cards from 52 deck of cards to obtain a three of a kind.

Now at first we choose the rank for which 3 cards have same rank. A full deck of playing card have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king and ace. For three of a kind, we choose 1 rank from this set of 13 ranks. The number of ways to get this

Now we choose the rank of nonmatching cards. There are 12 remaining ranks from these 12 ranks we choose two ranks. The number of ways to get this

Now we choose suits for three same card combination. In a full deck of cards there are four suits, from these four suits we choose three suits. The number of ways to get this

Now we choose a suit for one of the nonmatching cards. There are four suits, from these four suits we choose one suits. The number of ways to get this

Now we choose a suit for the other nonmatching card. There are four suits, from these we choose one suits. The number of ways to get this

Finally we compute the number of ways to get a three of a kind,

Number of ways to get a three of a kind is equal to product of each of this case.

i.e, Number of ways to get a three of a kind =

= 13×66×4×4×4 = 54912

Final Answer:- Number of ways to choose 5 cards from a 52 deck of cards to obtain three of a kind is = 54912