Chapter 15.7, Problem 22WE

To determine

To evaluate:

The value of the given number.

Answer to Problem 22WE

The value of given operation is $57,798$

Explanation of Solution

Given:

The order of operation

  $13-\text{card}$

Exactly $11\text{diamonds}$

Concept used:

Using by the formula of the permutation

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

Calculation:

There are $13$ diamonds in a deck of cards

The number of ways to choose $11$ diamonds is

  $\begin{array}{l}{}^{13}{C}_{11}=\frac{13!}{11!\left(13-11\right)!}\\ {}^{13}{C}_{11}=\frac{13\times 12\times 11!}{11!2!}\\ {}^{13}{C}_{11}=\frac{13\times 12}{2!}\\ {}^{13}{C}_{11}=13\times 6\\ {}^{13}{C}_{11}=78\end{array}$

Number of ways to pick $2$ other cards

  $\begin{array}{l}{}^{13}{C}_{11}=\frac{13!}{11!\left(13-11\right)!}\\ {}^{13}{C}_{11}=\frac{13\times 12\times 11!}{11!2!}\\ {}^{39}{C}_2=\frac{39\times 38}{2!}\\ {}^{39}{C}_2=39\times 19\\ {}^{39}{C}_2=741\end{array}$

The number of ways to have a $13-$ card hand with exactly $11$ diamonds is

  $\begin{array}{l}=78\times 741\\ =57,798\end{array}$

Therefore,

The value of given operation is $57,798$