Permutations Question Help

Question

1. from a deck of cards, 5 cards are chosen.

what is the likelihood of choosing 5 face cards

Answer 1

Without replacement: ₁₂C₅ / ₅₂C₅

_{With replacement (12/52)}⁵

Not "P"..."C"

₁₂C₅= 12!(5!*7!)=792

₅₂C₅₌ 52!/(5!*47!)=2598960

and the probability is 3.04*10^-4

The with replacement probability is 6.5*10^-4

Answer 2

In a standard deck of cards, there are 52 cards. There are 3 face cards in each suit (King, Queen, Jack) so there are 4*3 or 12 face cards.

There is no indication of how you are picking the 5 cards, so I will show you the solution for each.

Cards drawn and order maintained (Permutation):

The number of ways you can draw five face cards out of the 12 availably is: ₁₂P₅ = 12!/(12-5)! = 95040

The number of ways you can draw five cards (regardless of value) out of the 52 is: ₅₂P₅ = 52!/(52-5)! = 311875200

The likelihood of getting the 5 face cards is: 95040/311875200 ≈ 0.000305.

Cards drawn and simply grouped (Combination): (This is the more plausible interpretation of the question)

The number of ways you can draw five face cards out of the 12 availably is: ₁₂C₅ = 12!/((12-5)!5!) = 792

The number of ways you can draw five cards (regardless of value) out of the 52 is: ₅₂P₅ = 52!/((52-5)!5!) = 2598960

The likelihood of getting the 5 face cards is: 792/2598960 ≈ 0.000305.

(Interestingly, you get the same answer. That usually doesn't happen).