Combinations and Permutations in Committee Selection: A Comprehensive Guide

This article dives into the mathematical intricacies of selecting a committee of 5 people from a group of 6 men and 4 women with one specific individual required to be included. We explore the combinatorial methods and binomial coefficients to find the number of possible committees.

Introduction to Combinatorics

Combinatorics is a branch of mathematics that deals with the counting and arrangement of elements. It is particularly useful in solving problems related to selection and organization. Combinations and permutations are two core concepts in this field.

The Problem at Hand

A committee of 5 people is to be chosen from a group of 6 men and 4 women, with the condition that a specific person must be included in the committee. This problem requires the use of combinatorial mathematics to accurately determine the number of possible committees.

Solving the Problem

To solve this problem, we need to follow these steps:

Step 1: Choose the Required Person

Since one specific person must be included in the committee, we will consider this person already chosen. This leaves us with 4 more members to select from the remaining group.

Step 2: Determine the Remaining People

There are 5 people left to choose from after the required person is chosen. These include 5 men if the chosen person is a man or 4 women and 5 men if the chosen person is a woman.

Step 3: Calculate Possible Combinations

Using the combination formula, we can calculate the number of ways to choose 4 more members from the remaining 9 people:

binom{9}{4} frac{9!}{4!(9-4)!} frac{3024}{24} 126

In both scenarios (man or woman as the chosen person), we need to calculate binom{9}{4}, which gives us 126 possible committees.

Understanding the Binomial Coefficient

The binomial coefficient, binom{n}{k}, is defined as:

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

This formula is read as "n choose k" and represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.

Example Calculation

Let's break down the example calculations mentioned in the original problem:

4 Women Choose 1

The number of ways to choose 1 woman from 4 is:

binom{4}{1} 4

7 Men Choose 4

The number of ways to choose 4 men from 7 is:

binom{7}{4} 35

Total Combinations

The total number of ways to form a committee of 5 people with 1 woman is:

4 (women) × 35 (men) 140

This amount takes into account that there are 4 possible women and, for each woman, 35 ways to choose 4 men from 7 men.

Calculation Breakdown

Here’s the detailed breakdown for each scenario:

Scenario 1: Chosen Person is a Man

With 5 men and 4 women to choose from, we need to choose 4 more members from these 9 people:

binom{9}{4} 126

Scenario 2: Chosen Person is a Woman

With 6 men and 3 women to choose from, we again need to choose 4 more members from these 9 people:

binom{9}{4} 126

Therefore, the total number of possible committees, including the specified individual, is:

126 (man) 126 (woman) 252

Conclusion

In summary, by applying the principles of combinations and permutations, we have determined that there are 252 possible committees that can be formed under the given conditions.

Understanding these mathematical concepts is crucial in various real-world scenarios, such as organizing groups, drawing samples, and making selections. If you need further assistance with combinatorial problems or would like to delve deeper into the mathematical underpinnings of such problems, feel free to explore more resources on combinatorics and permutation theory.