Chapter 15.7, Problem 23WE

To determine

To evaluate:

The value of the given number.

Answer to Problem 23WE

The value of given operation is $4512$

Explanation of Solution

Given:

The order of operation

  $5-\text{card}$

Exactly $3\text{aces},2\text{other}\text{cards}$

Concept used:

Using by the formula of the permutation

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

Calculation:

There are $5$ diamonds in a deck of cards

The number of ways to choose $3$ aces is

  $\begin{array}{l}{}^4{C}_3=\frac{4!}{3!\left(4-3\right)!}\\ {}^4{C}_3=\frac{4\times 3!}{3!1!}\\ {}^4{C}_3=4\end{array}$

Number of ways to pick $2$ other cards

  $\begin{array}{l}{}^{48}{C}_2=\frac{48!}{2!\left(48-2\right)!}\\ {}^{48}{C}_2=\frac{48\times 47\times 46!}{2!46!}\\ {}^{39}{C}_2=\frac{47\times 48}{2!}\\ {}^{39}{C}_2=24\times 47\\ {}^{39}{C}_2=1128\end{array}$

The number of ways to have a $5-$ card hand with exactly $3$ aces and $2$ other card is

  $\begin{array}{l}=4\times 1128\\ =4512\end{array}$

Therefore,

The value of given operation is $4512$