Chapter 9.6, Problem 64E

a.

To determine

Find the numbers of different two-person relationships in groups of people.

Answer to Problem 64E

  ${}_3{C}_2=3$ .

Explanation of Solution

Calculation:

Consider the following question,

  ${}_3{C}_2$

You know that the number of permutations of elements taken at a time is,

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}\mathrm{..........}\left(1\right)$

Here,

  $\begin{array}{l}n=3\\ r=2\end{array}$

Put the values in the formula and solve as,

  $C{}_r=\frac{3!}{\left(3-2\right)!2!}$

  $\begin{array}{l}=\frac{3!}{1!2!}\\ =\frac{3\times 2\times 1}{1\times 2\times 1}\\ =3\end{array}$

Hence, the number of different two person relationships in groups of sizes ${}_3{C}_2=3$ .

b.

To determine

Find the numbers of different two-person relationships in groups of people

Answer to Problem 64E

  ${}_8{C}_2=28$

Explanation of Solution

Calculation:

Consider the following question,

  ${}_8{C}_2$

You know that the number of permutations of elements taken at a time is,

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}\mathrm{..........}\left(1\right)$

Here,

  $\begin{array}{l}n=8\\ r=2\end{array}$

Put the values in the formula and solve as,

  $C{}_r=\frac{8!}{\left(8-2\right)!2!}$

  $\begin{array}{l}=\frac{8!}{1!2!}\\ =\frac{8\times 7\times 6!}{6!\times 2\times 1}\\ =28\end{array}$

Hence, the number of different two person relationships in groups of sizes ${}_8{C}_2=28$ .

c.

To determine

Find the numbers of different two-person relationships in groups of people

Answer to Problem 64E

  ${}_{12}{C}_2=66$

Explanation of Solution

Calculation:

Consider the following question,

  ${}_{12}{C}_2$

You know that the number of permutations of elements taken at a time is,

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}\mathrm{..........}\left(1\right)$

Here,

  $\begin{array}{l}n=12\\ r=2\end{array}$

Put the values in the formula and solve as,

  $C{}_r=\frac{12!}{\left(12-2\right)!2!}$

  $\begin{array}{l}=\frac{12!}{1!2!}\\ =\frac{12\times 11\times 10!}{10!\times 2\times 1}\\ =66\end{array}$

Hence, the number of different two person relationships in groups of sizes ${}_{12}{C}_2=66$ .

d.

To determine

Find the numbers of different two-person relationships in groups of people

Answer to Problem 64E

  ${}_{20}{C}_2=190$

Explanation of Solution

Calculation:

Consider the following question,

  ${}_{20}{C}_2$

You know that the number of permutations of elements taken at a time is,

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}\mathrm{..........}\left(1\right)$

Here,

  $\begin{array}{l}n=20\\ r=2\end{array}$

Put the values in the formula and solve as,

  $C{}_r=\frac{20!}{\left(20-2\right)!2!}$

  $\begin{array}{l}=\frac{20!}{1!2!}\\ =\frac{20\times 19\times 18!}{19!\times 2\times 1}\\ =190\end{array}$

Hence, the number of different two person relationships in groups of sizes ${}_{20}{C}_2=190$ .