Question

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these 

(i) four cards are of the same suit 

(ii) four cards belong to four different suit 

(iii) are face cards 

(iv) two red cards and two are black card 

(v) 4 cards are of the same colour.

Answer 1

We have, 

Number of ways of choosing 4 cards from a pack of 52 cards is the number of combinations of 52 different things taken 4 at time 

∴ Required number of ways = ⁵²C₄

= (52 x 51 x 50 x 49)/(4 x 3 x 2 x 1) = 270725

(i) There are four suits, namely diamond, club, spade, heart and each suit has 13 cards. We have to choose 4 cards of the same suit so 4 diamond cards out of 13 diamond cards can be selected in ¹³C₄ ways. Similarly,there are ¹³C₄ ways of choosing 4 spades and ¹³C₄ ways of choosing 4 hearts. 

∴ Required number of ways 

= ¹³c₄ + ¹³c₄ + ¹³c₄ + ¹³c₄

= 4x¹³C₄ = 4 x (13 x 12 x 11 x 10)/(4 x 3 x 2 x 1) = 2860

(ii) There are 13 cards in each suit we have to select 4 cards belonging to 4 different suits. There are ¹³C₁ ways of choosing one card from 13 diamond cards, ¹³C₁ ways of choosing 1 card from 13 cards of spades, ¹³C₁ ways of choosing 1 card from 13 cards of club and ¹³C₁ ways of choosing 1 card from 13 cards of hearts. By multiplication principle, the required number of ways 

= ¹³C₁ x ¹³C₁ x ³C₁ x ³C₁ 

= 13 x 13 x 13 x 13 = 134  =28561

(iii) Face cards means Kings, Queen, Jack there are 4 suits, each suit has 3 face cards. Therefore there are 12 face cards and 4 are to be selected out of 12 cards, in 12C4 ways Required number of ways = 12C4

= (12 x 11 x 10 x 9)/(4 x 3 x 2 x 1) = 495

(iv) There are 26 red cards and 26 black cards. 

∴ We have select 2 red and 2 black cards. This can be done in ²⁶C₂ x ²⁶C₂ ways. 

= (26 x 25)/(2 x 1) x (26 x 25)/(2 x 1)

= 325 x 325 = 105625

∴ Required number of ways = ²⁶C₂ x ²⁶C₂

= (26 x 25)/(2 x 1) x (26 x 25)/(2 x 1)

= 325 x 325 = 105625

(v) There are 26 red and 26 black cards

∴ 4 red cards can be selected in ²⁶C₄ ways 4 black cards can be selected in ²⁶C₄ ways 

∴ Required number of ways = ²⁶C₄ + ²⁶C₄

= 2x²⁶C₄ = (26 x 25 x 24 x 26)/(4 x 3 x 2 x 1) = 29900