Combination Problems in Selecting Committees: A Comprehensive Analysis

When faced with the task of selecting a committee from a group of individuals, a fundamental mathematical concept is often employed: the combination. In this article, we will explore how to determine the number of ways to select a committee of 2 boys and 2 girls from a group of 4 boys and 3 girls using the combination formula. We will also analyze multiple approaches to solving this problem, ensuring a thorough understanding of the underlying principles.

Introduction to Combination

Combination is a fundamental concept in combinatorial mathematics and probability. It is used to determine the number of ways to choose a subset of items from a larger set, without regard to the order of selection. The combination formula is given by:

Combination Formula

The formula for the number of combinations is: binom{n}{r} frac{n!}{r!(n-r)!} (n) is the total number of items to choose from (r) is the number of items to choose ! denotes factorial (the product of all positive integers up to a specified number)

Problem Statement

The problem we are addressing is to form a committee consisting of 2 boys and 2 girls from a group of 4 boys and 3 girls. We will explore the solution using the combination formula and also provide an alternative method of solution.

Solution Using the Combination Formula

Step 1: Selecting the Boys

To determine the number of ways to choose 2 boys from 4, we use the combination formula:

binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6

Step 2: Selecting the Girls

Similarly, to determine the number of ways to choose 2 girls from 3, we again use the combination formula:

binom{3}{2} frac{3!}{2!(3-2)!} frac{3 times 2}{2 times 1} 3

Step 3: Calculating the Total Combinations

The total number of ways to form the committee is obtained by multiplying the number of ways to select the boys by the number of ways to select the girls:

text{Total ways} binom{4}{2} times binom{3}{2} 6 times 3 18

Thus, there are 18 possible ways to select the committee of 2 boys and 2 girls.

Alternative Method of Solution

Alternatively, we can determine the number of ways to select a subset of boys and girls through a different approach. Here are the steps:

Step 1: Selecting the Boys

We start by choosing 2 boys from 4. There are 12 possible combinations, as we can choose the first boy in 4 ways, the second in 3 ways (after selecting the first), but we have counted each combination twice (since the order of selection doesn’t matter). Thus, the number of unique combinations is:

frac{12}{2} 6

Step 2: Selecting the Girls

For the girls, we can choose 2 out of 3. Each choice of 2 girls corresponds to leaving one girl out. Hence, there are:

3 frac{3 times 2}{2 times 1}

Step 3: Calculating the Total Combinations

Each combination of the chosen boys can be paired with each combination of the chosen girls. Therefore, the total number of possible committees is:

6 times 3 18

This again results in 18 possible ways to select the committee.

The Final Answer

The total number of ways to select a committee of 2 boys and 2 girls from 4 boys and 3 girls is:

18

Conclusion

Understanding combinations is crucial in solving such problems. Whether using the combination formula directly or through alternative methods, the result remains consistent. By applying these principles, we can effectively tackle similar problems in a systematic and accurate manner.

Key Takeaways:

Combination formula: binom{n}{r} frac{n!}{r!(n-r)!} The order of selection does not matter in combinations. Different methods can be used to verify the result, ensuring accuracy.