# Thread: card deck word problem

1. ## card deck word problem

question:

we pull 6 cards from a standard deck of 52.

a) how many ways can we end up with all 6 being spades?

b) how many ways can we end up with 2 aces, 3 kings, and a queen?

if my notes are right (i don't think they are) part "a" would be 52 choose 6 and part "b" would be 52 choose 5.
can someone confirm or deny this please?

We pull 6 cards from a standard deck of 52.

a) How many ways can we end up with all 6 being Spades?

There are 13 Spades; we want 6 of them.

There are: . ${13\choose6} \:=\:\frac{13!}{6!\,7!} \:=\:1716$ ways.

b) How many ways can we end up with 2 Aces, 3 Kings, and a Queen?

$\begin{array}{ccc}\text{There are four Aces; we want two of them: } & {4\choose2} \:=\:6\text{ ways} \\ \\[-3mm]

\text{There are four Kings; we want three of them:} & {4\choose3} \:=\:4\text{ ways} \\ \\[-3mm]

\text{There are four Queens; we want one of them: } & {4\choose1} \:=\:4 \text{ ways} \end{array}$

Therefore, there are: . $6 \cdot 4 \cdot 4 \;=\;96$ ways.