Question

Someone is dealt a 6 card hand. Find the number of hands that fit in the following descriptions.

a) The cards are all red.

b) exactly four of the cards are clubs

c) Either three or four of the cards are face cards

d) There are at most four diamond cards in the hand

e) There are exactly two hearts and three diamonds in the hand

Answer 1

In a standard deck, there are 13 ordinal cards (Ace - 10, Jack, Queen, King) and in each of 4 suits (Hearts, Diamonds, Clubs, Spades) for a total of 13×4=52 cards.

Someone is dealt a 6 card hand.

a)

Total number of red cards in a deck =26

Therefore, the number of hands possible is

b)

Number of clubs card = 13

Number of cards which are not clubs = 52 - 13 = 39

We have to select 4 cards from 13 clubs and remaining 2 cards from 39 cards

So

Therefore, the number of hands possible = 529815

c)

Number of face cards =12

Number of cards which are not face cards = 52 -12 = 40

If three of the cards are face cards, then number of possible hands is

If four of the cards are face cards, then number of possible hands is

The number of possible hands is the sum of these two cases

2173600 + 386100

= 2559700

Therefore, the number of possible hands that either three or four of the cards are face cards = 2559700

d)

Number of diamond cards = 13

Number of cards which is not diamonds =52 -13 = 39

Number of ways to choose 6 cards with no diamond =  

Number of ways to choose 6 cards with 1 diamond =

Number of ways to choose 6 cards with 2 diamonds =

Number of ways to choose 6 cards with 3 diamonds =

Number of ways to choose 6 cards with 4 diamonds =

Number of possible hands that at most four diamond cards in the hand is

3262623 + 7484841 + 6415578 + 2613754 + 529815

= 20306611

Therefore, the number of possible hands at most four diamond cards in the hand = 20306611

e)

Number of hearts in 52 cards = 13

Number of diamonds in 52 cards =13

Number of cards which is not heart and diamond = 52 -(13+13) = 26

We have to select 2 cards from 13 hearts, 3 cards from 13 diamonds and remaining 1 card from 26 cards

So,

The number of possible hands is

Therefore, the number of possible hands that exactly two hearts and three diamonds in the hand = 580008