Problem 51 - Precalculus with Limits

Question

In Exercises 51 - 56, evaluate \( _nC_r \) using the formula from this section. \( _5C_2 \)

Step-by-Step Solution

Answer

The value of \( _5C_2 \) is 10.

Step 1: Calculating Factorials

Firstly, factorials need to be calculated. The factorial of a number is calculated by multiplying that number and all numbers below it down to 1. In this case, the factorials for n, r and (n-r) have to be calculated. For \( n = 5 \), \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). For \( r = 2 \), \( 2! = 2 \times 1 = 2 \). For \( n - r = 5 - 2 = 3 \), \( 3! = 3 \times 2 \times 1 = 6 \).

Step 2: Applying the Formula

Next, these factorial values are to be substituted into the original binomial coefficient formula to get the result. So, \( _5C_2 = \dfrac{5!}{2!(5-2)!} = \dfrac{120}{2 \times 6} = \dfrac{120}{12} = 10 \).

Key Concepts

Binomial Coefficient

Binomial coefficients are a key concept in combinatorics. They represent the number of ways to choose a subset of items from a larger set, without considering the order of selection. For any binomial coefficient, it's expressed as \( _nC_r \) or \( \binom{n}{r} \). Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to select.

The formula used to calculate this is given by:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

This formula essentially calculates the permutations of the selected items, divided by the permutations of the items not selected to prevent double-counting. By using this formula, we can find the number of possible ways to choose \( r \) items from \( n \) items, which is vital in determining probabilities and combinations in statistics and probability theory.

Factorials

Factorials are a fundamental part of various equations and are widely used in mathematics. A factorial, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, 5 factorial or \( 5! \) is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are crucial when calculating permutations and combinations because they help simplify the process of counting arrangements of items. In the calculation used for binomial coefficients, the factorials allow us to normalize the total permutations, ensuring that we only count the necessary arrangements or selections. This simplicity is why factorials are indispensable in combinatorial mathematics.

Permutations

Permutations refer to the arrangement of a set where the order does matter. When you calculate permutations, you're determining how many different ways you can order a set of items. This is crucial when the sequence is important, unlike combinations where the order is irrelevant.

The formula to determine permutations is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This formula finds the number of ways to arrange \( r \) items selected from \( n \) available items. Permutations are used in a variety of fields, including computer science, cryptography, and operations research. While the calculation of permutations looks similar to the binomial coefficient, they serve different purposes depending on whether or not the order of selection is important. Understanding the difference between permutations and combinations is vital, especially in solving complex problems involving arrangements and selections in probability theory.