Problem 15 - Algebra 2

Question

Evaluate each expression. \(C(5,2)\)

Step-by-Step Solution

Answer

The combination \( C(5, 2) \) is 10.

Step 1: Understanding the Combination Formula

To find the combination of 5 items taken 2 at a time, we use the combination formula:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]In this exercise, \( n = 5 \) and \( r = 2 \).

Step 2: Substitute Values into the Formula

Substitute \( n = 5 \) and \( r = 2 \) into the formula:\[ C(5, 2) = \frac{5!}{2!(5-2)!} \]This means we need to calculate \( 5! \), \( 2! \), and \( 3! \) (since \( 5-2 = 3 \)).

Step 3: Calculate Factorials

First, calculate the factorials:- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- \( 2! = 2 \times 1 = 2 \)- \( 3! = 3 \times 2 \times 1 = 6 \)

Step 4: Plug Factorials into the Combination Formula

Now, substitute the factorial values back into the combination formula:\[ C(5, 2) = \frac{120}{2 \times 6} \]

Step 5: Simplify the Expression

Simplify the fraction:\[ \frac{120}{2 \times 6} = \frac{120}{12} = 10 \]

Step 6: Final Answer

The number of combinations of 5 items taken 2 at a time is 10.

Key Concepts

Factorials

A factorial is an operation that you perform on a number and it is denoted by an exclamation point (!). The factorial of a number is the product of that number and all the positive integers below it.
For instance, when you see \( 5! \), it means 5 multiplied by all the numbers below it: \( 5 \times 4 \times 3 \times 2 \times 1 \). This results in 120. Factorials are fundamental in combinatorics, as they help us calculate permutations and combinations.
Here’s a quick guide on how to compute factorials:

• Start with the number you want the factorial of (denoted \( n \)).
• Multiply \( n \) by every number less than it down to 1.
• The resulting product is \( n! \).

Factorials grow very fast, which means the numbers get big quickly with even small increases in \( n \). They are essential for solving various problems that involve arrangements and selections.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and the counting of these arrangements. It's all about finding ways to organize a set of items. Whether you're arranging books on a shelf or choosing a subset of items, combinatorics provides the tools you need to calculate possible combinations efficiently.
In this field, you often deal with:

• Permutations: Arrangements where order matters.
• Combinations: Selections where order does not matter.

The key idea in combinatorics is understanding the difference between these concepts and applying the correct formula. It often involves using factorials because these allow you to organize and count these items quickly and systematically. Mastering combinatorics helps to solve real-world problems that involve grouping and arranging objects.

Combination Formula

The combination formula is a mathematical way to find out how many different groups can be selected from a set of items, where the order of selection does not matter. This formula is written as \( C(n, r) = \frac{n!}{r!(n-r)!} \) where:

• \( n \) is the total number of items.
• \( r \) is the number of items to be chosen.

To use this formula effectively, follow these steps:1. Calculate \( n! \), which is the factorial of the total number of items.
2. Calculate \( r! \), the factorial of the number of items to choose.
3. Calculate \((n-r)!\), the factorial of the difference between total items and chosen items.
4. Substitute these values into the formula and simplify.
For example, if you want to find out how many ways you can choose 2 items out of 5 (written as \( C(5, 2) \)), substitute into the formula to get \( \frac{5!}{2! \times 3!} \). After calculating the factorials and simplifying, you find there are 10 combinations.