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 there is exactly three aces in each combination.
Answer» 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

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

Answer»

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