Permutations And Combinations

Question

# What is the number of ways of choosing 4 cards from a pack of 52 cards? In how many of these :(i) four cards are of the same suit?(ii) four cards belongs to different suit?(iii) are face cards?(iv) two are red cards and two are black cards?(v) cards are of the same colour?

Solution

Number of cards in the deck = 52
Number of cards to be selected = 4
Number of selections =
= 270725
(i) The four cards are of the same suit.
Options are:
They are all club cards
Number of selections =
Or
Number of selections =
Or
They are all diamond cards.
Number of selections =
Or
They are all heart cards.
Number of selections =
Hence, the total number of selections =

=
(ii) They are one from each suit.
1 card is club and 1 card is spade and 1 card is diamond and 1 card is heart.
Total number of combinations =
= 13 x 13 x 13 x 13
= 28561.
(iii) The four cards are face cards.
Number of face cards = 4 kings + 4 queens + 4 jacks = 12
Number of other cards  =  52 - 12 = 40
4 out of 12 face cards are to be selected

Number of selections =
(iv) There are 26 black cards and 26 red cards.
Two are to be selected from black cards and two from the red cards.
Number of selections =
=

=  105625
Hence, the number of selections = 105625
(v) The four cards are of the same colour.
All four are either red or all four are black.
Number of selections =

=

=
= 29900.