What is the probability that a poker hand will have n kings

What is the probability that a poker hand will have n kings in it? (Answer for each possible value of n: 0, 1, 2, 3, or 4.) (5 card hand)

Suppose we play a game as follows: I deal you a poker hand, and then pay you $1 for each king in your hand. For instance, if your hand has 2 kings in it, you get $2. On average, how much money do you expect to win each time you play this game?

Solution

P [ 0 king from the 5 in my hand ] = (4 C 0)* ( 48 C 4) / ( 52 C 5) = 0.65884..
P [ 1 king from the 5 in hand ] = ( 4 C 1) * ( 48 C 4) / ( 52 C 5) = 0.29947..
P [ 2 king from the 5 in hand ] = ( 4 C 2 ) * ( 48 C 3 ) / ( 52 C 5 ) = 0.0399298

P [ 3 king from the 5 in hand ] = ( 4 C 3) * ( 48 C 2) / ( 52 C 5) = 0.001736079..

..
P [ 4 king from the 5 in hand ] = ( 4 C 4) * ( 48 C 1) / ( 52 C 5) = 0.0000185..


E( winning money ) = (1* 0.29947) + ( 2 * 0.0399298) + ( 3 * 0.001736079) + ( 4*0.0000185) = 0.3846118...