Short Question 7 & 8

Question 7:
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
$\begin{array}{l}={}^{10}{C}_5\\ =252\end{array}$

Question 8:
$\boxed{R}\boxed{E}\boxed{A}\boxed{C}\boxed{T}$
Solution:
(a) Number of arrangements = 5! = 120

(b) 
Step 1
If the letters ‘E ’ and ‘A’ have to be placed side by side, they will be considered as one item.
Together with the letters ‘R ’, ‘C ’ and ‘T ’, there are altogether 4 items.

Number of arrangements = 4!
The letters ‘E ’ and ‘A ’ can also be arranged among themselves in their group.
Number of arrangements = 2!

Hence, number of arrangements of all the letters of the word ‘REACT’ in which the letters E and A have to be side by side
= 4! × 2!
= 24 × 2
= 48