Question

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination.

Answer 1

No. of cards = 52 : In that number of aces = 4 

No. of cards needed = 5

In that 5 cards number of aces needed = 3 

So the 3 aces can be selected from 4 aces in ⁴C₃ = ³C₁ = 4 ways 

So the remaining = 5 – 3 = 2 

This 2 cards can be selected in ⁴⁸C₂ ways

⁴⁸C₂ = (48 x 47)/(2 x 1) = 1128 ways

∴ No. of ways in which the 5 cards can be selected = (⁴⁸C₂)(⁴C₃) 

= (48 x 47)/(2 x 1) = 4 = 4512