Committee Formation: Combinatorial Selection with Constraints

In combinatorial mathematics, determining the number of ways to select a committee from a group of individuals with specific constraints is a classic problem. One such problem involves forming a committee of five members from a group consisting of 4 girls and 7 boys, with certain conditions to satisfy. This article explores the different scenarios in which such a committee can be formed, ensuring a balance between boys and girls.

Committees with No Girls Allowed

Let us first consider the scenario where the committee consists only of boys. Given a group of 7 boys and the need to select a committee of 5, we use the combination formula to calculate the number of ways to form such a committee:

Step 1: Calculate the number of ways to choose 5 boys from 7 boys.

[ binom{7}{5} frac{7!}{5!(7-5)!} frac{7!}{5! cdot 2!} frac{7 times 6}{2 times 1} 21 text{ ways} ]

Committees with at Least 1 Boy and 1 Girl

The next scenario involves forming a committee with at least one boy and one girl. We explore all possible partitions of the committee that meet this criterion:

Case A: 1 boy 4 girls

[ binom{7}{1} cdot binom{4}{4} 7 cdot 1 7 text{ ways} ]

Case B: 2 boys 3 girls

[ binom{7}{2} cdot binom{4}{3} 21 cdot 4 84 text{ ways} ]

Case C: 3 boys 2 girls

[ binom{7}{3} cdot binom{4}{2} 35 cdot 6 210 text{ ways} ]

Case D: 4 boys 1 girl

[ binom{7}{4} cdot binom{4}{1} 35 cdot 4 140 text{ ways} ]

Adding all the cases together:

[ 7 84 210 140 461 text{ ways} ]

Thus, there are 461 different ways to form a committee with at least one boy and one girl.

Committees with at Least 3 Girls

Finally, let's consider the scenario where the committee must have at least three girls. We will again use the combination formula to find the number of valid committees.

Case E: 3 girls 2 boys

[ binom{4}{3} cdot binom{7}{2} 4 cdot 21 84 text{ ways} ]

Case F: 4 girls 1 boy

[ binom{4}{4} cdot binom{7}{1} 1 cdot 7 7 text{ ways} ]

Adding both cases together:

[ 84 7 91 text{ ways} ]

Conclusion

From the above calculations, we can summarize the different ways to form committees under various constraints. The key takeaway is that for a balanced and diverse committee, there are multiple methods to ensure the committee meets specific criteria, such as including at least one boy and one girl or having a minimum number of girls.

By understanding these combinatorial principles, one can effectively tackle similar problems involving committee formation and ensure that the selection process is both fair and inclusive.