Question

# A debate club consists of 6 girls and 4 boys. A team of 4 members is said to be selected from this club including the selection of captain (from among these 4 members) for the team. If the team has to be included at most one boy, then the number of ways of selecting the team is$A)380 \\ B)320 \\ C)260 \\ D)95 \\$

Hint: To solve this problem we need to have basic knowledge about combination concepts as the question is all about selecting the team members from given members. Here we should also know what does at most means as this is the condition that included in the question.

Given that in a debate club there are 6 girls and 4 boys.
So, here we have to select 4 members (from debate club) with at most 1 boy in a group.
Here the condition is at-most 1 boy which means 1 boy or less than 1 boy.
On the basis of data given we have 2possibilities to select 4 members with at most 1 boy in a team of 4.

First possibility:
All 4 members can be girls
This is possible because the condition is that we have to select 4 members with at-most 1 boy which means 1 or less than one.
Here we know that total number of girls in the debate club = 6
Here we have to select all 4 girls from 6 girls in the debate club.
Now on using the first possibility we can select 4 members in $^6{C_4} = 15ways$.

Second possibility:
3girls and 1 boy
Number of possibilities that we can select 4 member here =$^6{C_3}{ \times ^4}{C_1} = 80{\text{ways}}$
Number of ways that we can select 4 members =first possibility +second possibility
Number of ways that we can select 4 members = 15 ways+80ways = 90ways.

Therefore the number of ways of selecting team of 4 members including the selection of captain is $= 95 \times 4 = 380ways$
Therefore total number of required ways =380ways.

Note: In this problem we have to find the number of possible ways of selecting a team of 4 members using the condition at-most 1 boy in the group. In this problem we generally focus on the condition at-most and finish our answer but there is one more condition that we have to make a note of. That is the process of selection of a team of 4members includes the captain so captain should not be neglected.