Question

In a college a committee of 7 people has to be selected from a group of 8 fourth year and 6 third year students. In how many ways can this committee be selected if in the committee, majority of fourth year students is required?

• A. 2320
• B. 1960
• C. 1530
• D. 2216
• E. 2416

Answer 1

The correct option is E 2416

Soln: Majority of fourth year means out of 7, minimum 4 should be from fourth year.

Considering different possibilities:

$\begin{array}{ccccc}\text{No.from 4th year}& \text{Ways of choosing 4th year}& \text{No.from 3rd year}& \text{Ways of choosing 3rd year}& \text{wys of choosing commitee}\\ \text{4}& \text{8C4=70}& \text{3}& \text{6C3=20}& \text{70}\times \text{20 =1400}\\ \text{5}& \text{8C5=56}& \text{2}& \text{6C2=15}& \text{56}\times \text{15 =840}\\ \text{6}& \text{8C6=28}& \text{1}& \text{6C1=6}& \text{28}\times \text{6 =168}\\ \text{7}& \text{8C7=8}& \text{0}& \text{6C0=1}& \text{8}\times \text{1=8}\\ \text{Total}& & & & \text{2416}\end{array}$

So, total no. of ways = 2416. Hence option (E).