Forming a Committee of Three from Separate Departments: A Combinatorial Analysis

In many organizational contexts, it is necessary to form a committee consisting of members from different departments to ensure a balanced perspective. For example, a committee consisting of one member from the departments of statistics, mathematics, and chemistry needs to be formed. If 7 candidates are presented by each department, the question arises: how many different committees can be formed?

Principle of Multiplication in Combinatorics

The principle of multiplication in combinatorics states that if there are m ways to do one thing and n ways to do another, then there are m × n ways to do both. This principle can be extended to more than two events. Here, we will apply this principle to determine the number of different committees that can be formed.

Application of the Multiplication Principle

Given that 7 candidates are presented from each of the three departments (statistics, mathematics, and chemistry), we can calculate the total number of different committees as follows:

Step 1: Choosing a Member from Each Department

For the department of statistics, there are 7 possible candidates. For the department of mathematics, there are also 7 possible candidates. Similarly, for the department of chemistry, there are 7 possible candidates.

Step 2: Multiplication of Choices

To form a committee, we need to choose one member from each department. Using the multiplication principle, the total number of different committees can be calculated by multiplying the number of choices from each department:

Number of committees 7 (choices from statistics) × 7 (choices from mathematics) × 7 (choices from chemistry)

Mathematically, this can be expressed as:

73 343

Therefore, the total number of different committees that can be formed is 343.

Alternative Approach Using Combinatorial Principles

Another way to solve this problem is by considering the overall pool of candidates. There are a total of 21 candidates (7 from each of the three departments), making a total of 21 distinct candidates. We need to choose 3 members from these 21 candidates, ensuring that each member comes from a different department. This can be approached in the following manner:

Step 1: Combinatorial Selection

The number of combinations can be calculated by considering the permutations of selecting the candidates from different departments. The formula for permutations when n distinct objects are to be chosen in r distinct positions is given by P(n, r) n! / (n-r)!. In this case:

Total permutations 7! / (7-3)! 21 × 14 × 7

However, since the order in which we select the candidates does not matter, we need to divide by the factorial of the number of selections, which is 3! (3 factorial).

Therefore:

Total distinct committees (21 × 14 × 7) / 3! 2058 / 6 343

This confirms that the number of different committees that can be formed is indeed 343.

Conclusion

The process of forming a committee with one member from each of three distinct departments involves a straightforward application of the multiplication principle of combinatorics. Whether we approach it by considering individual selections or by using permutation principles, the result is the same: 343 different possible committees can be formed.

Key Takeaways

- The multiplication principle is a fundamental principle in combinatorics that helps in determining the number of possible outcomes when multiple choices are involved.

- Understanding and applying combinatorial principles is crucial in various scenarios, including committee formation, selection processes, and more.

- The calculation of 73 343 provides a clear and concise solution to the problem of forming a balanced committee with one representative from each of three departments.