Selecting a Committee of 5 Members from 10 People: A Comprehensive Guide

This article aims to explore the combinatorial problem of selecting a committee of 5 members from a group of 10 people, with the constraint that both the oldest and youngest individuals must be included. This is a fundamental problem in combinatorial mathematics and can be solved using various methods. We will explore these methods, explain the steps involved, and discuss the underlying principles.

Understanding Combinatorial Problems

A combinatorial problem is a type of counting problem that deals with the selection, arrangement, and operation of elements in sets. The classic problem we are addressing here is to determine how many ways a committee of 5 members can be selected from a group of 10 people when certain members must be included. This problem can be represented mathematically using the concept of combinations.

The Formula for Combinations

The formula for the number of combinations of n items taken k at a time is given by:

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

Where:

! represents the factorial of a number, n is the total number of items, k is the number of items to be selected.

In our case, we have n 10 and k 5.

The Step-by-Step Solution

Using the formula, we can calculate the number of ways to form the committee:

C105 frac{10!}{5!(10-5)!}

Breaking down the factorial:

C105 frac{10 times 9 times 8 times 7 times 6}{5 times 4 times 3 times 2 times 1}

Performing the arithmetic, we get:

C105 252

Alternative Methods for Calculation

There are alternative ways to solve this problem that do not require direct calculation of factorials:

Pascal's Triangle

The coefficients of combinations can also be found in Pascal's Triangle. For n 10, the 5th row (starting from 0) provides the combination value, which is 252.

Permutations and Arrangements

Another method involves calculating permutations of the remaining members. If we select the oldest and youngest members first, we have 8 remaining members to choose from for the other 3 spots. The number of ways to select 3 members from 8 is:

8 times 7 times 6 / (1 times 2 times 3) 56

Conclusion and Practical Applications

Understanding how to calculate such combinations is valuable. These skills go beyond mere academic exercises and can be practical in various real-world scenarios, such as team selection, scheduling, and resource allocation.

If you want to perform these calculations on a calculator, such as a TI-84 or TI-83 online app, you can use the nCr function to find the number of combinations quickly and accurately. This skill is especially useful when working in fields that require data analysis and decision-making, such as finance, operations management, and computer science.

By understanding these methods, you will be better equipped to tackle similar combinatorial problems in the future, even without the aid of a calculator.