Please show work Five cards are drawn at random from an ordi

Please show work.

Five cards are drawn at random from an ordinary deck of 52 cards, that the 5-card hand contains the following: 3 kings and 2 queens No diamonds 3 cards of the same suit At least 3 aces

Solution

a)

n(picking any 5 card from deck) = 52 C 5

n( 3 Kings, 2 Queens) = 4 C 3 * 4 C 2

P( 3 Kings, 2 Queens ) = n( 3 Kings, 2 Queens) / n(picking any 5 card from deck) = 4 C 3 * 4 C 2 / 52 C 5 = nCr(4,3)*nCr(4,2)/nCr(52,5) =  0.00000923

b)

n( other than diamond picking 5) = 39 C 5

P(other than diamond picking 5 ) = n(other than diamond picking 5) / n(picking any 5 card from deck) = 39 C 5 / 52 C 5 = nCr(39,5)/nCr(52,5) =  0.22153

c)

n( 3 cards of same suit) = n( 3 clubs, 2 remaining) + n(3 hearts, 2 remaining) + n(3 spade, 2 remaining) + n(3 diamonds, 2 remaining) = 4 * 13 C 3 * 39 C 2

P( 3 cards of the same suit) = 4 * 13 C 3 * 39 C 2 / 52 C 5 = 4*nCr(13,3)nCr(39,2)/nCr(52,5) = 0.32617

d)

n(atleast 3 aces) = n( 3 aces, 2 remaining) + n( 4 aces, 1 remaining) = 4 C 3 * 48 C 2 + 4 C 4 * 48 C 1

P(atleast 3 aces) = [ 4 C 3 * 48 C 2 + 4 C 4 * 48 C 1] / 52 C 5 = [nCr(4,3)nCr(48,2)+nCr(4,4)nCr(48,1)]/nCr(52,5) = 0.0017545