Chapter 6.1, Problem 16E

Solution

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

  $2,574\text{ hands}$

Given information:

The cards should be: 5 hearts or 5 diamonds

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 13 cards of the four suits (spades, clubs, hearts and diamonds).

5 Hearts:

The number of ways to choose 5 hearts from these 13 heart cards is: $C{}_5$

Use the above formula for finding combinations value:

  $\begin{array}{c}C{}_5=\frac{13!}{\left(13-5\right)!\cdot 5!}\\ =\frac{13!}{8!\cdot 5!}\\ =\frac{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot \cancel{8!}}{\cancel{8!}\cdot \left(5\cdot 4\cdot 3\cdot 2\cdot 1\right)}\\ =1287\text{ hands}\end{array}$

5 Diamonds:

The number of ways to choose 5 diamonds from the 13 diamond cards is: $C{}_5$

Use the above formula for finding combinations value:

  $\begin{array}{c}C{}_5=\frac{13!}{\left(13-5\right)!\cdot 5!}\\ =\frac{13!}{8!\cdot 5!}\\ =\frac{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot \cancel{8!}}{\cancel{8!}\cdot \left(5\cdot 4\cdot 3\cdot 2\cdot 1\right)}\\ =1287\text{ hands}\end{array}$

So, total number of possible hands is:

  $1287+1287=2,574\text{ hands}$