Chapter 4.4, Problem 13E

a.

To determine

The value of 8!

Answer to Problem 13E

The value of 8! is40,320.

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, 8! is calculated as follows:

$\begin{array}{c}8!=8\times \left(8-1\right)\left(8-2\right)\times \cdot \cdot \cdot \times 1\\ =8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =40,320\end{array}$

Thus, the value of 8! is40,320.

b.

To determine

The value of 10!

Answer to Problem 13E

The value of 10! Is3,628,800 .

Explanation of Solution

Calculation:

10! is calculated as follows:

$\begin{array}{c}10!=10\left(10-1\right)\left(10-2\right)\cdot \cdot \cdot 1\\ =10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =3,628,800\end{array}$

Thus, the value of 10! is362,880.

c.

To determine

The value of 0!

Answer to Problem 13E

The value of 0! is1.

Explanation of Solution

Calculation:

By, using the fact,

$0!=1$

Thus, the value of 0! is1.

d.

To determine

The value of 1!

Answer to Problem 13E

The value of 1! is1.

Explanation of Solution

Calculation:

1! is calculated as follows:

$1!=1$

Thus, the value of 1! is1.

e.

To determine

The value of ${}_7{P}_5$

Answer to Problem 13E

The value of ${}_7{P}_5$ is 2,520.

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 5

$\begin{array}{c}{}_7{P}_5=\frac{7!}{\left(7-5\right)!}\\ =\frac{7!}{2!}\\ =\frac{7\times 6\times 5\times 4\times 3\times \cancel{2}\times \cancel{1}}{\cancel{2}\times \cancel{1}}\\ =2,520\end{array}$

Thus, the value of ${}_7{P}_5$ is 2,520.

f.

To determine

The value of ${}_{12}{P}_4$.

Answer to Problem 13E

The value of ${}_{12}{P}_4$ is 11,880.

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

$\begin{array}{c}{}_{12}{P}_4=\frac{12!}{\left(12-4\right)!}\\ =\frac{12!}{8!}\\ =\frac{12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}\\ =11,880\end{array}$

Thus, the value of ${}_{12}{P}_4$ is 11,880.

g.

To determine

The value of ${}_5{P}_3$

Answer to Problem 13E

The value of ${}_5{P}_3$ is 60.

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 5 and r as 3

$\begin{array}{c}{}_5{P}_3=\frac{5!}{\left(5-3\right)!}\\ =\frac{5!}{2!}\\ =\frac{5\times 4\times 3\times 2\times 1}{2\times 1}\\ =60\end{array}$

Thus, the value of ${}_5{P}_3$ is 60.

h.

To determine

The value of ${}_6{P}_0$.

Answer to Problem 13E

The value of ${}_6{P}_0$ 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 6 and r as 0

$\begin{array}{c}{}_6{P}_0=\frac{6!}{\left(6-0\right)!}\\ =\frac{6!}{6!}\\ =\frac{\cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{6}\times \cancel{5}\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =1\end{array}$

Thus, the value of ${}_6{P}_0$ is 1.

i.

To determine

The value of ${}_5{P}_5$

Answer to Problem 13E

The value of ${}_5{P}_5$ is 120.

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 5 and r as 5

$\begin{array}{c}{}_5{P}_5=\frac{5!}{\left(5-5\right)!}\\ =\frac{5!}{0!}\\ =\frac{5\times 4\times 3\times 2\times 1}{1}\\ =120\end{array}$

Thus, the value of ${}_5{P}_5$ is 120.

j.

To determine

The value of ${}_6{P}_2$

Answer to Problem 13E

The value of ${}_6{P}_2$ is 30.

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 2

$\begin{array}{c}{}_6{P}_2=\frac{6!}{\left(6-2\right)!}\\ =\frac{6!}{4!}\\ =\frac{6\times 5\times \cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}{\cancel{4}\times \cancel{3}\times \cancel{2}\times \cancel{1}}\\ =30\end{array}$

Thus, the value of ${}_6{P}_2$ is 30.