Determine the number of 5-card combinations out of a deck of 52 cards

1.	Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king?
Answer» We have to select one king out of 4 kings and four other cards out of 48. This can be done in ⁴C, x ⁴⁸C₄ ways = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3)/(2 x 1) = 907200 ∴ Required number of ways = ⁴C₁ x ⁴⁸C₄ = 4 x 2 x 47 x 46 x 45 = 778320

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king?

Answer»

We have to select one king out of 4 kings and four other cards out of 48. This can be done in ⁴C, x ⁴⁸C₄ ways

= (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3)/(2 x 1) = 907200

∴ Required number of ways = ⁴C₁ x ⁴⁸C₄

= 4 x 2 x 47 x 46 x 45 = 778320