Combinations and Committees: Calculating Different Group Formations

In various situations, such as forming committees or selecting groups from a pool of people, understanding the principles of combinations is vital. This article will focus on a specific problem where we need to form a committee of a certain size with specific constraints, and the mathematical methods to achieve it.

Problem Statement

We have a group of 5 men and 5 women. We need to form a committee of size 3, with 2 men and 1 woman, and we must include a specific woman in the committee. Our task is to determine the number of different committees that can be formed under these conditions.

Step-by-Step Solution

Select the Woman

Since one specific woman must be in the committee, we start by considering her as already chosen. This leaves us with the task of selecting 2 men from the group of 5 men.

Select the Men

To determine the number of ways to choose 2 men from 5, we use the combination formula:

( binom{n}{r} frac{n!}{r!(n-r)!} )

Here, ( n 5 ) and ( r 2 ).

Plugging in the values, we get:

( binom{5}{2} frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10 )

Conclusion

Since the specific woman is already included in the committee, the total number of committees of size 3 that can be formed with 2 men and 1 woman, including the specific woman, is 10. Therefore, the final answer is ( 10 ) committees.

Additional Scenarios

No Restrictions

If there are no restrictions, we can calculate the number of ways to choose 3 people out of 9:

( binom{9}{3} frac{9!}{3!(9-3)!} frac{9 times 8 times 7}{1 times 2 times 3} 84 )

1 Man and 2 Women

There are 4 men and 5 women. We can choose 1 man out of 4 in 4 ways and 2 women out of 5 in 10 ways. Therefore, the number of ways to form a committee with 1 man and 2 women is:

( 4 times 10 40 )

1 Fixed Woman, 1 Man, and 1 Woman

If one specific woman is fixed, we need to choose 1 man out of 4 and 1 woman out of the remaining 4 (5 - 1). The number of ways to do this is:

( 4 times 5 20 )

1 Fixed Man and 1 Man, 1 Woman

If one specific man is fixed, we need to choose 1 man out of the remaining 3 (5 - 1) and 1 woman out of 5. The number of ways to do this is:

( 3 times 5 15 )

Conclusion

Understanding combinations and committees is crucial for solving problems involving group selections. By using the combination formula and considering specific conditions, we can determine the number of different committees that can be formed under various constraints.