**Question 5**:

A committee that consists of 6 members is to be selected from 5 teachers and 4 students. Find the number of different committees that can be formed if

**(a)**there is no restriction,

**(b)**the number of teachers must exceed the number of students.

*Solution:***(a)**

Total number of committees = 5 + 4 = 9

6 members to be selected from 9 committees with no restriction

${}^{\mathrm{=\; 9}}{C}_{6}=84$

**(b)**

$\begin{array}{l}\text{Ifthenumberofteachersmustexceedthe}\\ \text{numberofstudents,thecombination}\\ \text{=4teachers2students+5teachers1student}\\ ={\text{}}^{5}{C}_{4}\times {\text{}}^{4}{C}_{2}+{\text{}}^{5}{C}_{5}\times {\text{}}^{4}{C}_{1}\\ =30+4\\ =34\end{array}$

**Question 6**:

Six members of a committee of a school are to be selected from 6 male teachers, 4 female teachers and a male principal. Find the number of different committees that can be formed if

**(a)**the principal is the chairman of the committee,

**(b)**there are exactly 2 females in the committee,

**(c)**there are not more than 4 males in the committee.

*Solution:***(a)**

If the principal is the chairman of the committee, the remaining number of committee is

**5**members.

Hence, the number of different committees that can be formed from the remaining 6 male teachers and 4 female teachers

$\begin{array}{l}={\text{}}^{10}{C}_{5}\\ =252\end{array}$

**(b)**

$\begin{array}{l}\text{Exactly2femalesinthecommittee}\\ {\text{=}}^{4}{C}_{2}\times {\text{}}^{7}{C}_{4}\\ =210\end{array}$

**(c)**

$\begin{array}{l}\text{Therearenotmorethan4malesinthecommittee}\\ \text{=4males2females+3males3females+2males4females}\\ ={\text{}}^{7}{C}_{4}\times {\text{}}^{4}{C}_{2}+{\text{}}^{7}{C}_{3}\times {\text{}}^{4}{C}_{3}+{\text{}}^{7}{C}_{2}\times {\text{}}^{4}{C}_{4}\\ =210+140+21\\ =371\end{array}$

**Question 7**:

A school prefect committee that consists of 6 persons is to be chosen from 6 Malays, 5 Chinese and 4 Indians. Calculate the number of different committees that can be formed if the number of Malays, Chinese and Indians must be equal.

*Solution:*Number of different committees that can be formed for 2 Malays, 2 Chinese and 2 Indians

$$\begin{array}{l}={\text{}}^{6}{C}_{2}\times {\text{}}^{5}{C}_{2}\text{}\times {\text{}}^{4}{C}_{2}\\ =900\end{array}$$

**Question 8**:

There are 10 different flavour candies in a plastic bag.

Find

**(a)**the number of ways 3 candies can be chosen from the plastic bag.

**(b)**the number of ways at least 8 candies can be chosen from the plastic bag.

*Solution:***(a)**

Number of ways choosing 3 candies out of 10 candies

$\begin{array}{l}={\text{}}^{10}{C}_{3}\\ =120\end{array}$

**(b)**

Number of ways choosing 8 candies $={\text{}}^{10}{C}_{8}$

Number of ways choosing 9 candies = ${}^{10}{C}_{9}$

Number of ways choosing 10 candies = ${}^{10}{C}_{10}$

Hence, number of ways of choosing at least 8 candies

$\begin{array}{l}={\text{}}^{10}{C}_{8}{\text{+}}^{10}{C}_{9}+{\text{}}^{10}{C}_{10}\\ =56\end{array}$