Question

A standard 52-card poker deck consists of 4 suits and 13 ranks. In how many ways can you draw 5 cards so that

1) there are no constraints?

2)all 5 cards are of same suits?

3) all four suits are present?

4) all cards are of distinct ranks?

Answer 1

1) number of ways to draw 5 cards from 52 cards =⁵²C₅ =2598960

2) all 5 cards are of same suits =N(all 5 from 1st suit)+N(all 5 from 2nd suit)+N(all 5 from 3rd suit)+N(all 5 from 4th suit)

=¹³C₅+¹³C₅+¹³C₅+¹³C₅=1287+1287+1287+1287 =5148

3)

all four suits are present =N(2 from first suit and 1 from each of remaining)+N(2 from 2nd suit and 1 from each of remaining)+N(2 from 3rd suit and 1 from each of remaining)+N(2 from 4th suit and 1 from each of remaining)

=¹³C₂*¹³C₁*¹³C₁¹³C₁ +¹³C₂*¹³C₁*¹³C₁¹³C₁ +¹³C₂*¹³C₁*¹³C₁¹³C₁ +¹³C₂*¹³C₁*¹³C₁¹³C₁

=685464

4) all cards are of distinct ranks =N(selecting five suits from 13 and then select one card from 4 from each of rank)

=¹³C₄*4*4*4*4*4 =1317888