Chapter 6.1, Problem 13E

Solution

To find: the number of possible 5-card hands that contains the cards specified.

  $729$

Given information:

The cards should be: 5 face cards (kings, queens and jacks)

These cards are taken from a standard 52-card deck.

Formula Used:

Combinations of n Objects Taken r at a Time

The number of combinations of r objects taken from a group of n distinct objects is denoted by $C{}_r$ and is given by the formula:

  $C{}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

Some known values are:

  $\begin{array}{l}1!=1\\ 0!=1\end{array}$

Explanation:

It is known that there are 3 face cards (king, queen, jack) in each of the four suits (spades, clubs, hearts and diamonds).

So that's $3\times 4=12$ face cards out of 52 cards.

Thus, the number of possible 5-card hands that contains 5 face cards is given by:

  $C{}_5$

Use the above formula for finding combinations value:

  $\begin{array}{c}C{}_5=\frac{12!}{\left(12-5\right)!\cdot 5!}\\ =\frac{12!}{7!\cdot 5!}\\ =\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot \cancel{7!}}{\cancel{7!}\cdot \left(5\cdot 4\cdot 3\cdot 2\cdot 1\right)}\\ =729\end{array}$