 # 4.4.2 Permutation Short Questions (Question 5 – 8)

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
${}^{= 9}{C}_{6}=84$

(b)
$\begin{array}{l}\text{If the number of teachers must exceed the}\\ \text{number of students, the combination}\\ \text{= 4 teachers 2 students + 5 teachers 1 student}\\ ={\text{}}^{5}{C}_{4}×{\text{}}^{4}{C}_{2}+{\text{}}^{5}{C}_{5}×{\text{}}^{4}{C}_{1}\\ =30+4\\ =34\end{array}$

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)
$\begin{array}{l}\text{Exactly 2 females in the committee}\\ {\text{=}}^{4}{C}_{2}×{\text{}}^{7}{C}_{4}\\ =210\end{array}$

(c)
$\begin{array}{l}\text{There are not more than 4 males in the committee}\\ \text{= 4 males 2 females + 3 males 3 females + 2 males 4 females}\\ ={\text{}}^{7}{C}_{4}×{\text{}}^{4}{C}_{2}+{\text{}}^{7}{C}_{3}×{\text{}}^{4}{C}_{3}+{\text{}}^{7}{C}_{2}×{\text{}}^{4}{C}_{4}\\ =210+140+21\\ =371\end{array}$

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

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