Question

Out of $6$ boys and $4$ girls, a group of $7$ is to be formed. In how many ways can this be done, if the group is to have a majority of boys?

• A. $120$
• B. $80$
• C. $90$
• D. $100$

Answer 1

The correct option is D

$100$

Explanation for the correct answer:

Finding the number of ways:

Number of boys $=6$

Number of girls $=4$

Number of members in the group$=7$

A group of $7$ members with the majority of boys can be formed in the following ways
Case i: $4$ boys and $3$ girls
Case ii: $5$ boys and $2$ girls
Case iii: $6$ boys and $1$ girl

Case i: choosing $4$ boys out of $6$$={}^6{C}_4$

choosing $3$ girls out of is $4$$={}^{4}C_3$

Total ways $={}^6{C}_4$ $\times$ ${}^{4}C_3$ $=60$ ways

Case ii: choosing $5$ boys out of is $6$$={}^6{C}_5$

choosing $2$ girls out of is $4$ $={}^{4}C_2$

Total ways $={}^6{C}_5$ $\times$ ${}^{4}C_2$$=36$ ways

Case iii: choosing $6$ boys out of is $6$ $=1$ way

choosing $1$ girl out of is $4$ $={}^{4}C_1$

Total ways $=1$ $\times$ ${}^{4}C_1$ $=4$ ways

Total possible combinations $=60+36+4=100$

Hence, option (D) is the correct answer