Problem 21 - Calculus for Biology and Medicine

Question

A committee of 3 people must be formed from a group of 10. How many committees can there be if no specific tasks are assigned to the members?

Step-by-Step Solution

Answer

There are 120 different committees possible.

Step 1: Understand the Problem

We need to find out how many different ways we can select a group of 3 people from a total of 10 without considering the order of selection. This is a combination problem.

Step 2: Identify the Formula for Combinations

We use the combination formula for selecting r items from n items. The formula is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Apply the Values to the Formula

Set \( n = 10 \) and \( r = 3 \) in the formula. Therefore, we apply:\[ C(10, 3) = \frac{10!}{3!(10-3)!} \] This simplifies to \( \frac{10!}{3!7!} \).

Step 4: Simplify the Factorial Expression

Calculate the factorial expressions reducing the fraction step by step:- 10! = 10 × 9 × 8 × 7!- 3! = 3 × 2 × 1- 7! at the denominator will cancel out with 7! in the numerator.This simplifies the expression to: \[ \frac{10 \times 9 \times 8}{3 \times 2 \times 1} \]

Step 5: Perform the Calculation

Calculate the numerator and denominator:- Numerator: 10 × 9 × 8 = 720- Denominator: 3 × 2 × 1 = 6Divide the numerator by denominator:\[ \frac{720}{6} = 120 \]

Step 6: Conclusion

There are 120 different ways to form a committee of 3 people from a group of 10.

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics that help determine the number of ways to arrange items. The factorial of a number, denoted as \( n! \), represents the product of all positive integers from 1 up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). This mathematical operation simplifies complex calculations, particularly those involving permutations and combinations.

When calculating combinations, factorials are used to simplify expressions, such as \( \frac{10!}{3!7!} \) in our example, by canceling common factors in the numerator and the denominator. This makes the process of counting possible arrangements manageable and accurate, ensuring we don't overcount or undercount possibilities.

Binomial Coefficient

The binomial coefficient is a key element in combinatorics and is denoted as \( C(n, r) \) or \( \binom{n}{r} \). It represents the number of ways to choose \( r \) items from \( n \) items without considering the order, making it essential for tackling combination problems such as forming a committee from a larger group.

The formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) gives a precise count of the selections. In our example, \( C(10, 3) \) computes the number of ways to choose 3 people out of 10, which results in 120. This coefficient is widely used not only in combinatorics but also in probability and algebra, playing a vital role in binomial theorem applications.

Combinatorics

Combinatorics is the branch of mathematics that focuses on counting, arranging, and analyzing different configurations of objects. It provides methods for solving problems involving selections and arrangements, ensuring a comprehensive count of possibilities, whether considering order or not.

In the given problem, combinatorics allows us to determine the total number of ways to form a committee by using combination calculations. By considering only the selection without regard to order, combinatorics helps simplify real-world problems, making it applicable in fields such as computer science, statistics, and decision-making processes. Understanding these principles equips learners with tools to solve complex problems effectively.