Find the probability with 5-card hands from a deck of 52 cards that contains.. A) 3 aces and 2 kings B) 3 queens and 2 spades?

A)
Number of combinations to choose 5 cards from 52 cards without restriction
= C(52,5)

To choose 3 aces and 2 kings :
• Choose 3 aces from 4 aces (C(4,3)), and
• Choose 2 kings from 4 kings (C(4,2))
Number of combinations to choose 3 aces and 2 kings
= C(4,3) × C(4,2)

P(3 aces and 2 kings)
= C(4,3) × C(4,2) / C(52,5)
= 4 × 6 / 2598960
= 1/108290


B)
There are 2 cases to choose 3 queens and 2 spades.

Case I : with spade queen
Choose the spade queen (C(1,1)), 2 queens from the rest 3 queens (C(3,2)) and 2 spades from the rest 12 spades (C(12, 2)).
Number of combinations of Case I
= C(1,1) × C(3,2) × C(12, 2)
= C(3,2) × C(12, 2)

Case II : without spade queen
Choose the club queen (C(1,1)), diamond queen(C(1,1)), heart queen (C(1,1)) and 2 spades from the 13 spades (C(13,2)).
Number of combinations of Case II
= C(1,1) × C(1,1) × C(1,1) × C(13, 2)
= C(13, 2)

P(3 queens and 2 spades)
= [C(3,2) × C(12, 2) + C(13, 2)] / C(52,5)
= [3 × 66 + 78] / 2598960
= 23/216580

B/
with Q spade --->(3C2)(12C2)/ (52C5)
without Q spade ---> (3C3)(12C2)/ (52C5)

P(3 Q and 2 Spades but not Q spade) = (3C2)(12C2)/ (52C5) + (3C3)(12C2)/ (52C5)

4C3 = 4
4C2 = 6
52C5 = 2,598,960

(4C3∙4C2)/52C5 = 9.23446E-06 = 1/108,290