Problem 62 - Precalculus with Limits

Question

In order to conduct an experiment, five students are randomly selected from a class of \( 20 \). How many different groups of five students are possible?

Step-by-Step Solution

Answer

The number of different groups of five students that can be selected from a class of twenty is \( C(20, 5) \)

Step 1: Understand the Combination Formula

The combination formula is used to find the number of ways a certain number of items can be chosen from a larger number without considering the order. It is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.

Step 2: Apply the Values to the Combination Formula

In this case, the total number of students \( n \) is 20, and the number of students to be chosen \( r \) is 5. Substituting these values into the combination formula gives us \( C(20, 5) = \frac{20!}{5!(20-5)!} \)

Step 3: Calculate the Factorials

Factorial of a number is the product of all positive integers up to that number. So, calculate the factorials of 20, 5 and 20-5 which is 15.

Step 4: Solve the Equation

Divide the factorial of 20 by the product of the factorials of 5 and 15 to get the number of ways the students can be selected.

Key Concepts

Combination Formula

The combination formula is a powerful tool in combinatorics, helping determine the number of ways to choose items from a larger set without caring about the order. This is in contrast to permutations, where order does matter. The formula for combinations is given by:\[C(n, r) = \frac{n!}{r!(n-r)!}\]Here, \( n \) represents the total number of things to choose from, while \( r \) is the number of things to be chosen. The "!" symbol is called a factorial and is key in calculating combinations. With combinations, it’s crucial to remember that the order of selection doesn’t influence the outcome. For example, selecting students to form a committee doesn’t matter in which order they are picked, just who is on the committee.
When applying this in real situations as with our exercise, you decide the number from which to choose (20 students) and how many to select (5 students) to find the total number of combinations possible.

Factorials

Factorials are fundamental components in combinatorics and are used extensively in the combination formula as well. The factorial of a number \( n \) (written as \( n! \)) is the product of all positive integers less than or equal to \( n \). This means that:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 15! = 15 \times 14 \times \ldots \times 2 \times 1 \)
• \( 20! = 20 \times 19 \times \ldots \times 2 \times 1 \)

Factorials grow very quickly. Thus, calculations often involve simplifications. In our combination problem, simplifying by dividing the largest possible common terms in the numerator and denominator, like \( 20! \) divided by \( 5! \times 15! \), reduces computation. Understanding factorial behavior helps handle large numbers without direct computation, vital for accurate results.

Permutations

Permutations may sound similar to combinations, but the key difference is that order does matter. If you're arranging books on a shelf or deciding the lineup for a race, each arrangement or order is distinct and counts differently. The formula for permutations of \( n \) items taken \( r \) at a time is:\[P(n, r) = \frac{n!}{(n-r)!}\]Unlike combinations, where order is irrelevant, permutations require each different sequence to be unique. Imagine lining up the same five students. The order in which they stand would create different permutations.

• Example: for 3 books, choosing 2 gives permutations of AB, BA, AC, CA, BC, CB.

While our exercise used combinations, knowing permutations can help in situations where sequence and position influence the outcome. Both concepts provide a foundational understanding of how to count possibilities accurately.