Problem 82 - Algebra 2

Question

Evaluate each expression. \(_{5} \mathrm{C}_{5}\)

Step-by-Step Solution

Answer

The evaluated expression for \(_{5} \mathrm{C}_{5}\) is 1.

Step 1: Understand the notation

The notation \(_nC_r\) is read as 'n choose r' which is the number of combinations possible when choosing r elements from a group of n elements, regardless of the order.

Step 2: Use the formula

In the given exercise, we need to evaluate \(_{5}C_{5}\), i.e., when n = 5 and r = 5. We can use the combination formula, \(_nC_r = \frac{n!}{r!(n-r)!}\), and substituting the values, we get \( \frac{5!}{5!(5-5)!}\).

Step 3: Calculate factorial

The factorial of a number is the product of an integer and all the integers below it until 1. In our case, 5! = 5 × 4 × 3 × 2 × 1 = 120 and 0! = 1. Note, by definition 0! is equal to 1.

Step 4: Substitute and simplify

Substituting the values of the factorials computed, we get \( \frac{120}{120 * 1}\). This simplifies to 1.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics, where the goal is to determine how many ways we can select a subset of items from a larger set, without caring about the order. Imagine you have a bag of 5 marbles, and you want to know how many different ways you can select all 5 of them at once. This is precisely what combinations help us figure out.

• The term "combination" indicates that the order of selection does not matter.
• We use a special notation for combinations: \(_nC_r\), pronounced as "n choose r", where "n" is the total number of items, and "r" is the number of items to choose.

It's important to recognize that combinations are key when the sequence doesn't matter, as opposed to permutations, which account for order. Every combination problem can be solved by using the same formula and logical approach, making them predictable and straightforward once you understand the basics.

Factorial Calculation

In mathematical equations, particularly in combinations and permutations, you'll often encounter factorials. A factorial, denoted by an exclamation mark (!), is a product of an integer and all the integers below it, down to 1.For example, calculating \(5!\) involves multiplying all whole numbers from 5 down to 1:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are an integral part of calculating combinations because they help determine how different ways items can be arranged. A special case is \(0!\), which by definition is 1. This might seem a bit odd at first, but it's a necessary rule to ensure that all mathematical models using factorials work correctly, including our combination formula.Understanding how to compute factorials reliably will greatly aid in solving combinatorial problems, because any error in this basic calculation will lead to incorrect results further in your problem-solving journey.

Binomial Coefficient

The binomial coefficient is an elegant tool in combinatorics that helps us understand combinations in a formulaic way. It's represented as \(_nC_r\), which signifies choosing "r" items from "n" without regard to the order of selection.The formula for the binomial coefficient is:\[_nC_r = \frac{n!}{r!(n-r)!}\]This means that to compute how many ways we can choose "r" items from "n", you take the factorial of "n", divide it by the factorial of "r" and the factorial of the difference between "n" and "r".For example, when solving \(_5C_5\), you set "n" and "r" to 5, and your formula becomes:

• \(\frac{5!}{5!(5-5)!} = \frac{120}{120 \, \cdot \, 1} = 1\)

This results in 1, signifying there is only one way to choose all elements from the set, reflecting the intuitive idea that if you want to select all items, there's just one combination: selecting them all. Understanding and applying the binomial coefficient enables you to tackle a wide variety of problems in combinatorics efficiently.