Question

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

1. four cards are of the same suit,

2. are face cards,

3. two are red cards and two are black cards,

4. cards are of the same colour? 

Answer 1

Selection of 4 cards from 52 = 52C₄

1. There are 4 suits in a pack of 52 playing cards. They are Club, Spade, Diamond, and Heart. Selecting 4 from each can be done in,

= 13C₄ + 13C₄ + 13C₄ + 13C₄

= 4 × 13C₄ = 4 × \(\frac{13 \times 12 \times 11 \times 10}{1 \times 2 \times 3 \times 4}\) = 2860.

2. There are 12 face cards in a pack of 52 playing cards. Selection 4 cards can be done in 12C₄ = 495.

3. There are 26 red cards and 26 black in a pack of 52 playing cards. Selection of 2 cards should be done from each colour, this can be done in 26C₂ × 26C₂ = (325)² = 105625.

4. Selection of 4 cards from same colour = 26C₄ + 26C₄ = 29900