Probability Probelm: Can you please help me with this problem?

Question

A poker hand consists of 5 cards from a standard deck of 52. Find the number of different poker hands of the specified type. two pairs (two of one denomination, two of another denomination, and one of a third)

For those unfamiliar with playing cards, here is a short description. A standard deck consists of 52 playing cards. Each card is in one of 13 denominations: ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, jack (J), queen (Q), and king (K), and in one of four suits: hearts (♥), diamonds (♦), clubs (♣), and spades (♠). Thus, for instance, the jack of spades, J♠, refers to the denomination of jack in the suit of spades. The entire deck of cards is as shown below.

Answer 1

In a hand with two pair you will see three different denominations --

two of them will provide a pair each, one of them will provide a single card.

First we figure out all the possible ways of choosing the denominations,

and then for each choice of denominations we count the number of way

of choosing the suits.

Since this platform does not allow me to enter the normal notation for

"n chose k", I will write it as (n,k).

A = (13,2) is the number of ways of choosing the two denominations of pairs.

For each such choice there are

B = 11 ways of choosing the denomination of a single.

For each such choice there are

C = 4 choices of a suit for the denomination of singles.

For each such choice there are

D = (4,2) ways of choosing a pair of suits for one pair.

For each such choice there are

E = (4,2) ways of choosing a pair of suits for the other pair.

Evaluating:

A = (13,2) = 13 x 12 / 2 = 78

D = E = (4,2) = 4 x 3 / 2 = 6

So the total number of such poker hands is

A x B x C x D x E = 78 x 11 x 4 x 6 x 6 = 123,552