Question

Five cards are drawn from a pack of 52 well-shuffled cards. Find the probability that the 4 cards are aces AND 1 card is a king. 

Answer 1

There are 4 aces and 4 kings in a standard deck of 52 cards.

Then

$$P(4\ Aces\ \&\ 1\ King)=\dfrac{\dbinom{4}{4}\dbinom{4}{1}}{\dbinom{52}{5}}$$

$$=\dfrac{1(4)}{\dfrac{52!}{5!(52-5)!}}=\dfrac{4(120)}{52(51)(50)(49)(48)}$$

$$=\dfrac{1}{649740}\approx0.00000154$$