Chapter 11.6, Problem 40E

a.

To determine

To calculate: the $P{}_1$ .

Answer to Problem 40E

  $P{}_1=5$

Explanation of Solution

Given :

  $P{}_1,P{}_2,P{}_3,P{}_4,P{}_5$

Formula used:

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

Calculation : to find all the expressions put the value in the given formula,

  $\begin{array}{l}P{}_1=\frac{5!}{\left(5-1\right)!}\text{ [substitue the n by 5 and r by 1, subtract and divide out the common factors]}\\ P{}_1=5\end{array}$

it is needed to look for the pattern t in permutations s of n letters. A pattern t is said to occur in a permutation s if there are integers 1 = i1 < i2 < ...

Hence, $P{}_1=5$

b.

To determine

To calculate: the $P{}_2$ .

Answer to Problem 40E

  $P{}_2=20$

Explanation of Solution

Given :

  $P{}_1,P{}_2,P{}_3,P{}_4,P{}_5$

Formula used:

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

Calculation : to find the expressions put the value in the given formula,

  $\begin{array}{l}P{}_2=\frac{5!}{\left(5-2\right)!}\text{ [substitue the n by 5 and r by 1, subtract and divide out the common factors}]\\ P{}_2=20\end{array}$

it is needed to look for the pattern t in permutations s of n letters. A pattern t is said to occur in a permutation s if there are integers 1 = i1 < i2 < ...

Hence, $P{}_2=20$

c.

To determine

To calculate: the $P{}_3$ .

Answer to Problem 40E

  $P{}_3=60$

Explanation of Solution

Given :

  $P{}_1,P{}_2,P{}_3,P{}_4,P{}_5$

Formula used:

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

Calculation : to find the expressions put the value in the given formula,

  $\begin{array}{l}P{}_3=\frac{5!}{\left(5-3\right)!}\text{ [substitue the n by 5 and r by 1, subtract and divide out the common factors}]\\ P{}_3=60\end{array}$

it is needed to look for the pattern t in permutations s of n letters. A pattern t is said to occur in a permutation s if there are integers 1 = i1 < i2 < ...

Hence, $P{}_3=60$

d.

To determine

To calculate: the $P{}_4$ .

Answer to Problem 40E

  $P{}_4=120$

Explanation of Solution

Given :

  $P{}_1,P{}_2,P{}_3,P{}_4,P{}_5$

Formula used:

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

Calculation : to find the expressions put the value in the given formula,

  $\begin{array}{l}P{}_4=\frac{5!}{\left(5-4\right)!}\text{ [substitue the n by 5 and r by 1, subtract and divide out the common factors}]\\ P{}_4=120\end{array}$

it is needed to look for the pattern t in permutations s of n letters. A pattern t is said to occur in a permutation s if there are integers 1 = i1 < i2 < ...

Hence, $P{}_4=120$

e.

To determine

To calculate: the $P{}_5$ .

Answer to Problem 40E

  $P{}_5=120$

Explanation of Solution

Given :

  $P{}_1,P{}_2,P{}_3,P{}_4,P{}_5$

Formula used:

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

Calculation : to find the expressions put the value in the given formula,

  $\begin{array}{l}P{}_5=\frac{5!}{\left(5-5\right)!}\text{ [substitue the n by 5 and r by 1, subtract and divide out the common factors}]\\ P{}_5=120\end{array}$

It is needed to look for the pattern t in permutations s of n letters. A pattern t is said to occur in a permutation s if there are integers 1 = i1 < i2 < ...

Hence, $P{}_5=120$ .