Chapter 12.5, Problem 43SR

a.

To determine

To explain does the situation represent permutation or combination.

Answer to Problem 43SR

The situation represents combination.

Explanation of Solution

Given information:

The millers have twelve grandchildren, 5boys and 7 girls.

The situation of selecting 3 grandchildren for the entire 12 membered group represents combination because in this situation the order is not important where in case of permutation the order is important.

Use definition of permutation and combination.

b.

To determine

To determine the number of ways to choose three members out of twelve members.

Answer to Problem 43SR

Number of ways to select three members out of twelve members is 220.

Explanation of Solution

Given information:

The millers have twelve grandchildren, 5boys and 7 girls.

The number of ways to choose three members out of twelve members using combination formula.

  $\begin{array}{l}C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}\\ C\left(12,3\right)=\frac{12!}{\left(12-3\right)!3!}\\ C\left(12,3\right)=\frac{12\cdot 11\cdot 10\cdot 9!}{9!\cdot 3!}\\ C\left(12,3\right)=\frac{12\cdot 11\cdot 10}{3!}\\ C\left(12,3\right)=\frac{1320}{6}\\ C\left(12,3\right)=220\end{array}$

Therefore,

Number of ways to select three members out of twelve members is 220.

c.

To determine

To calculate the probability that all three will be girls.

Answer to Problem 43SR

Probability of all three selected members will be girls $=\frac{35}{220}or15.9\%$

Explanation of Solution

Given information:

The millers have twelve grandchildren, 5boys and 7 girls.

The probability of all three selected members will be girls.

The no ways to select three girls

  $\begin{array}{l}C\left(n,r\right)=\frac{n!}{\left(n-r\right)!\cdot r!}\\ C\left(7,3\right)=\frac{7!}{\left(7-3\right)!\cdot 3!}\\ C\left(7,3\right)=\frac{7\cdot 6\cdot 5\cdot 4!}{4!\cdot 3!}\\ C\left(7,3\right)=\frac{7\cdot 6\cdot 5}{3!}\\ C\left(7,3\right)=\frac{210}{6}\\ C\left(7,3\right)=35\end{array}$

Therefore,

Probability of all three selected members will be girlsw $=\frac{35}{220}or15.9\%$