Question

Consider a 5-card poker hand.

A) How many hands are there with exactly 3 Aces that is NOT a Full House.

B) How many hands with two pairs (but no 3 of a kind)

Answer 1

(A)

Step 1:

Total cards in hand = 5

Step 2:

3 Aces can be selected from 4 Aces in:

ways

Step 3:

Remaining 2 cards are not Aces and the 2 cards can be selected from remaining 48 cards in:

ways

So,

Total number of hands= 4 X 1128 = 4512.

So,

Answer is:

4512

(B)

To find hands with 2 pairs:

Step 1:

Select the 2 distinct suits out of the total 13 suits for 2 pairs. This can be done in:

ways

Step 2:

Select 2 cards out of the 4 cards from each of the 2 suits selected. This can be done in:

Step 3:

Select the suit for the 5th card from the remaining 13 - 2 = 11 suits. This can be done in:

ways

Step 4: Select 1 card out of the 4 cards in the suit selected. This can be done in:

ways.

Thus, total number of hands = 78 X 36 X 11 X 4 = 123,552