Problem 24 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$C(16,3)$$

Step-by-Step Solution

Answer

The value of \( C(16, 3) \) is 560.

Step 1: Understand the combination formula

The combination formula is used to determine how many ways you can choose a subset of items from a larger set, where the order does not matter. The formula is \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

Step 2: Substitute values into the formula

Substitute \( n = 16 \) and \( k = 3 \) into the combination formula: \( C(16, 3) = \frac{16!}{3!(16-3)!} \).

Step 3: Simplify the denominator

Simplify the denominator by evaluating \( 16 - 3 = 13 \), so the formula becomes \( C(16, 3) = \frac{16!}{3! \times 13!} \).

Step 4: Calculate factorials

Calculate the factorials needed: \( 16! = 16 \times 15 \times 14 \times 13! \) and \( 3! = 3 \times 2 \times 1 \). Notice that \( 13! \) in the numerator and denominator will cancel out.

Step 5: Simplify and calculate

Cancel \( 13! \) from both the numerator and denominator. The expression becomes \( \frac{16 \times 15 \times 14}{3 \times 2 \times 1} \). Evaluate the expression: \( 16 \times 15 = 240 \), \( 240 \times 14 = 3360 \). Divide by \( 3 \times 2 \times 1 = 6 \) to get \( 560 \).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics often used in counting problems. A factorial of a positive integer \[ n \] is denoted by \( n! \) and is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
Factorials provide a way to calculate the total number of permutations or arrangements of a set. Importantly, \( 0! \) is defined to be 1 because there is only one way to arrange a set of zero objects—which is to do nothing! Factorials grow very fast as \( n \) increases and are key in calculations involving combinations and permutations. When calculating combinations, factorials help by organizing and canceling terms to find efficient, simplified results.

Combination Formula

The combination formula is a tool used in mathematics to find how many ways you can select items from a larger group without regard to order. This formula is \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. This formula helps answer questions such as: "How many ways can I select 3 apples from a basket of 16 apples?" Here's a brief breakdown of how the formula works.

• **\( n! \)**: Calculates all possible arrangements of your total items.
• **\( k! \)**: Removes the permutations within the chosen items, because the order does not matter.
• **\( (n-k)! \)**: Removes the permutations in the unchosen items, ensuring that repeats are not counted.

When you substitute values into this formula as shown in the exercise, you can simply work through each part to find the number of combinations.

Mathematical Expressions

Mathematical expressions like those used in the combination formula can seem daunting at first, but breaking them down into manageable steps helps simplify the process. Mathematical expressions are combinations of numbers, variables, operators (like +, -, ×, ÷), and grouping symbols (like parentheses). When solving, it's important to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right). In the case of the combination formula, you first resolve the factorial parts before proceeding to division:

• Evaluate factorials, which might involve multiplying series of decreasing positive numbers.
• Cancel common factors in the numerator and denominator for simplicity.
• Select proper calculation operations step-by-step to reach the final result.

Understanding and accurately translating these mathematical concepts ensures successful problem solving in diverse scenarios.