Chapter 10.5, Problem 77E

(a)

To determine

The different 5-card hands possible in 52 playing cards.

Answer to Problem 77E

The possible five card hands are $2598960$

Explanation of Solution

Given:

standard deck of 52 playing cards.

Calculation:

For $r\le n$ , the number of combinations of n objects taken r at a time is given by the following expression.

  $C{}_r=\frac{n!}{(n-r)!r!}$ .......(1)

The total cards available are 52. Out of these cards the possible five card hands are $C{}_5$ .

From the eqn (1) let evaluate $C{}_5$ as below.

  $C{}_5=\frac{52!}{5!(52-5)!}$

  $\begin{array}{l}C{}_5=\frac{52\times 51\times 50\times 49\times 48\times 47!}{(5\times 4\times 3\times 2\times 1)\times 47!}\\ C{}_5=\frac{52\times 51\times 50\times 49\times 48}{(5\times 4\times 3\times 2\times 1)}\\ C{}_5=2598960\end{array}$

Hence, the possible five card hands are $2598960$

Conclusion:

Therefore, the possible five card hands are $2598960$

(b)

To determine

The different 5-card hands have all five cards.

Answer to Problem 77E

The number of five cards hands that have all five cards of a single suit is 5148

Explanation of Solution

Calculation:

One single suit consists of 13 cards. Out of these 13 cards the possible 5 card combinations are $C{}_5$ . According to the equation solve $C{}_5$ .

  $C{}_r=\frac{n!}{(n-r)!r!}$ .......(1)

  $\begin{array}{l}C{}_5=\frac{13!}{5!(13-5)!}\\ C{}_5=\frac{13!}{5!\times 8!}\\ C{}_5=\frac{13\times 12\times 11\times 10\times 9\times 8!}{(5\times 4\times 3\times 2\times 1)8!}\\ C{}_5=1287\end{array}$

Out off four suit the number of possibilities to pick a single suit is $C{}_1$ .according to equation(1)

  $\begin{array}{l}C{}_1=\frac{4!}{1!(4-1)!}\\ C{}_1=\frac{4!}{3!}\\ C{}_1=\frac{4\times 3!}{3!}\\ C{}_1=4\end{array}$

Hence, the number of five cards hands that have all five cards of a single suit is,

  $\begin{array}{l}C{}_5.C{}_1=1287\times 4\\ C{}_5.C{}_1=5148\end{array}$

Hence, the number of five cards hands that have all five cards of a single suit is $\boxed{5148}$ .

Conclusion:

Therefore, the number of five cards hands that have all five cards of a single suit is 5148.