# Team of 3 children, & a captain is required. 10 children can play chess of whom only 3 qualified to act captain. Find number of ways to select team.

Question : A team of three children plus a captain is required for a school chess competition. A particular school has ten children who can play chess of whom only three are qualified to act as captain.
In how many ways can a team be selected assuming that the three children qualified as captain are also eligible to play chess?

My Try: $$(^3C_1\cdot\; ^7C_3) + (^3C_2 \cdot \; ^7C_2) + (\;^3C_3 \cdot \; ^7C_1) = 175$$

Claimed Answer : The answer is $$252$$, however, I was not able to reach here. If Anyone know how to solve this question, please guide.

• What answer are you getting, and how did you arrive at it ? May 28 at 7:25
• I only got 3C1*7C3 + 3C2 * 7C2 + 3C3 * 7C1 = 175
– yuyu
May 28 at 7:27
• Once you choose one of the three eligible children as team captain, the other nine can still fill the other three team positions, with order being unimportant $\ (252 \ = \ 3 · 84) \ \ .$ May 28 at 8:06

Hint

You have forgotten that the $$3$$ who can become captains can also play chess. If you correct for that, you should get the right answer.

The simplest way is

[Choose captain]*[Choose $$3$$ more from all 9 remaining]

$$\dbinom31\dbinom93 = 252$$

• I already included them inside the group of 4?
– yuyu
May 28 at 7:53
• Since a long answer has been posted, I'm posting a short method May 28 at 9:03

Case 1: Choosing 1 captain from 3 eligible captains and 3 players from 7 children (who aren't eligible to be captain).

Number of teams $$= \binom{3}{1} * \binom{7}{3}$$

Case 2: Choosing 2 captains from 3 eligible captains and 2 players from 7 students. For every selection of children we have to choose a captain of the team, which means that every selection can form two possible with different captains.

Number of teams $$= 2 * \binom{3}{2} * \binom{7}{2}$$

Case 3: Choosing 3 captains from 3 eligible captains and 1 player from 7 children. This time we can choose the captain from 3 eligible children.

Number of teams $$= 3 * \binom{3}{3} * \binom{7}{1}$$

Therefore the total number of ways of forming a team

$$= \binom{3}{1} * \binom{7}{3} + 2 * \binom{3}{2} * \binom{7}{2} + 3 * \binom{3}{3} * \binom{7}{1} = 252$$