Chapter 4.4, Problem 13E

Evaluate each expression.

a. 11!

b. 9!

c. 0!

d. 1!

e. ₆P₄

f. ₁₂P₈

g. ₇P₇

h. ₄P₀

i. ₉P₂

j. ₁₁P₃

To determine

The value of 11!

Answer to Problem 13E

The value of 11! is 39,916,800.

Explanation of Solution

Calculation:

The formula used to find the factorial of any number n is given below:

$\begin{array}{l}n!=n\left(n-1\right)\left(n-2\right)\mathrm{...}\left(1\right)\\ 0!=1\end{array}$

Thus, 11! is calculated as follows:

$\begin{array}{c}11!=11\left(11-1\right)\left(11-2\right)\times \cdot \cdot \cdot \times 1\\ =11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =39,916,800\end{array}$

Thus, the value of 11! is 39,916,800.

To determine

The value of 9!

Answer to Problem 13E

The value of 9! is 362,880.

Explanation of Solution

Calculation:

9! is calculated as follows:

$\begin{array}{c}9!=9\left(9-1\right)\left(9-2\right)\cdot \cdot \cdot 1\\ =9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =362,880\end{array}$

Thus, the value of 9! is 362,880.

To determine

The value of 0!

Answer to Problem 13E

The value of 0! is 1.

Explanation of Solution

Calculation:

By, using the fact,

$0!=1$

Thus, the value of 0! is 1.

To determine

The value of 1!

Answer to Problem 13E

The value of 1! is 1.

Explanation of Solution

Calculation:

1! is calculated as follows:

$1!=1$

Thus, the value of 1! is 1.

To determine

The value of ${}_6{P}_4$

Answer to Problem 13E

The value of ${}_6{P}_4$ is 360.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 6 and r as 4

$\begin{array}{c}{}_6{P}_4=\frac{6!}{\left(6-4\right)!}\\ =\frac{6!}{2!}\\ =\frac{6\times 5\times 4\times 3\times \cancel{2}\times \cancel{1}}{\cancel{2}\times \cancel{1}}\\ =360\end{array}$

Thus, the value of ${}_6{P}_4$ is 360.

To determine

The value of ${}_{12}{P}_8$

Answer to Problem 13E

The value of ${}_{12}{P}_8$ is 19,958,400.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 12 and r as 8

$\begin{array}{c}{}_{12}{P}_8=\frac{12!}{\left(12-8\right)!}\\ =\frac{12!}{4!}\\ =\frac{12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =19,958,400\end{array}$

Thus, the value of ${}_{12}{P}_8$ is 19,958,400.

To determine

The value of ${}_7{P}_7$

Answer to Problem 13E

The value of ${}_7{P}_7$ is 5,040.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 7 and r as 7

$\begin{array}{c}{}_7{P}_7=\frac{7!}{\left(7-7\right)!}\\ =\frac{7!}{0!}\\ =\frac{7\times 6\times 5\times 4\times 3\times 2\times \cancel{1}}{\cancel{1}}\\ =5,040\end{array}$

Thus, the value of ${}_7{P}_7$ is 5,040.

To determine

The value of ${}_4{P}_0$

Answer to Problem 13E

The value of ${}_4{P}_0$ is 1.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 4 and r as 0

$\begin{array}{c}{}_4{P}_0=\frac{4!}{\left(4-0\right)!}\\ =\frac{4!}{4!}\\ =\frac{\cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =1\end{array}$

Thus, the value of ${}_4{P}_0$ is 1.

To determine

The value of ${}_9{P}_2$

Answer to Problem 13E

The value of ${}_9{P}_2$ is 72.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 9 and r as 2

$\begin{array}{c}{}_9{P}_2=\frac{9!}{\left(9-2\right)!}\\ =\frac{9!}{7!}\\ =\frac{9\times 8\times \cancel{7}\times \cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{7}\times \cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =72\end{array}$

Thus, the value of ${}_9{P}_2$ is 72.

To determine

The value of ${}_{11}{P}_3$

Answer to Problem 13E

The value of ${}_{11}{P}_3$ is 990.

Explanation of Solution

Calculation:

Permutation:

A permutation is an arrangement of n objects in r ways in a specific order and is given by,

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute n as 11 and r as 3

$\begin{array}{c}{}_{11}{P}_3=\frac{11!}{\left(11-3\right)!}\\ =\frac{11!}{8!}\\ =\frac{11\times 10\times 9\times \cancel{8}\times \cancel{7}\times \cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{8}\times \cancel{7}\times \cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =990\end{array}$

Thus, the value of ${}_{11}{P}_3$ is 990