Problem 58 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$C(5,3)$$

Step-by-Step Solution

Answer

10

Step 1: Understand the Combination Formula

To evaluate the combination \(C(n, k)\), use the formula: \[C(n, k) = \frac{n!}{k!(n - k)!}\]

Step 2: Identify Values for n and k

For \(C(5,3)\), identify \(n = 5\) and \(k = 3\).

Step 3: Calculate the Factorials

Calculate the factorials: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\), \(3! = 3 \times 2 \times 1 = 6\), and \(2! = 2 \times 1 = 2\)

Step 4: Substitute into the Formula

Substitute these values into the combination formula: \[C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{120}{6 \times 2}\]

Step 5: Simplify the Expression

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

Key Concepts

combination formula

Combinations are a fundamental concept in algebra. They allow us to determine the number of ways to select items from a larger group. The combination formula, denoted as \(C(n, k)\), helps us find this number:
\[C(n, k) = \frac{n!}{k!(n - k)!}\] Here, \(n\) is the total number of items, and \(k\) is the number of items to be selected. This formula ensures that order does not matter in our selection. For example, selecting 3 students from a group of 5 can be calculated using this formula.

factorial calculation

Factorials are essential in calculating combinations. The factorial of a number \(n\), written as \(n!\), is the product of all positive integers up to that number. For instance:
\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\] Factorials grow quickly, so it's helpful to understand the calculations step-by-step. Understanding factorials is crucial for simplifying expressions in combination problems.

binomial coefficient

The binomial coefficient, often written as \( \binom{n}{k} \), is another way to represent combinations. It is equal to \(C(n, k)\) and is used in the binomial theorem, which expands expressions raised to a power. For example, the coefficient \( \binom{5}{3} \) can be calculated as:
\[\binom{5}{3} = \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = 10 \] The binomial coefficient helps simplify polynomial expansions and other algebraic problems involving combinations.