Chapter 6.1, Problem 14E

Solution

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

  $48$

Given information:

The cards should be: 4 kings and 1 other card

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 is 1 king in each of the four suits (spades, clubs, hearts and diamonds).

So that's 4 king cards out of 52 cards.

Now, the remaining 1 card to be chosen can be any of the $\left(52-4=48\right)$ cards.

Thus, the number of possible 5-card hands that contains 4 kings and 1 other cards is given by:

  $C{}_4\times C{}_1$

Use the above formula for finding combinations value:

  $\begin{array}{c}C{}_4\times C{}_1=\left[\frac{4!}{\left(4-4\right)!\cdot 4!}\right]\left[\frac{48!}{\left(48-1\right)!\cdot 1!}\right]\\ =\left[\frac{4!}{0!\cdot 4!}\right]\left[\frac{48!}{47!\cdot 1}\right]\\ =\left[\frac{\cancel{4!}}{1\cdot \cancel{4!}}\right]\left[\frac{48\cdot \cancel{47!}}{\cancel{47!}\cdot 1}\right]\\ =1\cdot 48\\ =48\end{array}$