Problem 57 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$C(5,2)$$

Step-by-Step Solution

Answer

The value is 10.

Step 1: Understand the Binomial Coefficient Formula

The binomial coefficient, denoted as \(C(n, k)\), is calculated as \(\frac{n!}{k!(n-k)!}\). In this exercise, \(n = 5\) and \(k = 2\).

Step 2: Calculate the Factorials

Calculate the factorials needed for the formula: \(5!\), \(2!\), and \(3!\) (since \(3 = 5 - 2\)). \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\), \(2! = 2 \times 1 = 2\), and \(3! = 3 \times 2 \times 1 = 6\).

Step 3: Substitute Factorials into the Formula

Substitute the factorials into the binomial coefficient formula: \(C(5,2) = \frac{5!}{2!(5-2)!}\). This becomes \(C(5,2) = \frac{120}{2 \times 6}\).

Step 4: Simplify the Expression

Simplify the expression: \(C(5,2) = \frac{120}{12} = 10\).

Key Concepts

Factorials

Factorials play a crucial role in the calculation of binomial coefficients. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For example, the factorial of 4 is calculated as: 4! = 4 × 3 × 2 × 1 = 24. The factorial of 0 is defined as 1. Understanding factorials is important because they simplify the process of working with large numbers in combinatorics.

In the given exercise, we need to find the factorials of 5, 2, and 3 to solve for the binomial coefficient. Calculating these factorials involves multiplying a series of descending natural numbers. This basic operation forms the backbone of more complex computations in permutations and combinations.

Combinatorics

Combinatorics is a field of mathematics that deals with counting, arrangement, and combination of objects. It is fundamental in fields such as probability and statistics. The binomial coefficient is an essential concept in combinatorics, used to determine the number of ways to choose a subset of items from a larger set.

The binomial coefficient, expressed as \(C(n, k)\), represents the number of ways to select k items from a set of n items without regard to the order of selection. In the exercise, \(C(5,2)\) indicates how many ways we can choose 2 items from 5. The formula for binomial coefficients is \(\frac{n!}{k!(n-k)!}\). By applying this formula, you can find the number of possible combinations.

Mathematical Expressions

Mathematical expressions involve numbers, variables, and operators arranged in a meaningful way. They are used to represent real-world scenarios mathematically. In the context of binomial coefficients, mathematical expressions are used to simplify and solve combinatorial problems.

For instance, in the step-by-step solution provided, we start with the general formula for the binomial coefficient: \(C(n, k) = \frac{n!}{k!(n-k)!}\). By substituting the values of n and k, and then calculating and simplifying the factorials, we break down a complex problem into manageable parts. Simplifying mathematical expressions allows us to solve problems accurately and efficiently. This methodical approach is critical for understanding and applying mathematical concepts in various disciplines.