Four cards are dealt from a standard deck of 52 cards. How many ways can you be dealt one jack and three queens?
There are how many ways to be dealt one jack and three queens
Given that four cards are dealt from a standard deck of 52 cards.
We have to find the number of ways in which one can be dealt one jack and three queens.
A standard deck of 52 cards have 2 red jack and 2 black jack.
Hence, the total number of jacks in a standard deck of 52 cards is 4.
Similarly, since there are 2 red queens and 2 black queens, the total number of queens in a standard deck of 52 cards is 4.
The number of ways of dealing with one jack is given by the number of ways of selecting one jack from the 4 jacks available.
We know that number of ways of selecting r items from n items is given by $n{C}_{r}$.
Hence, the number of ways of selecting one jack from the 4 jacks = $4{C}_{1}=\frac{4!}{1!\times \left(4-1\right)!}=\frac{4!}{3!}=4$.
Similarly, the number of ways of dealing with three queens is given by the number of ways of selecting three queens from the 4 queens available = $4{C}_{3}=\frac{4!}{3!\times \left(4-3\right)!}=\frac{4!}{3!}=4$.
Note that, the event of choosing one jack is independent of the event of choosing three queens as removal of a jack does not affect the number of queens and the removal of queens do not affect the number of jacks.
We know that if A and B are 2 independent events such event A can be done in a ways and event B can be done in b ways, then A and B can be done in ab ways.
Here, the event of dealing with one jack and the event of dealing with three queens are independent events with the number of ways of dealing with one jack = 4 and the number of ways of dealing with three queens = 4.
Hence, the total number of ways in which one can be dealt one jack and three queens is given by $4\times 4=16$.