Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, compute the value of the expression. $$ C(8,5) $$

Step-by-Step Solution

Answer

The value of \( C(8, 5) \) is 56.

Step 1: Understand the Notation

The expression \( C(n, k) \) represents a combination. It computes the number of ways to choose \( k \) items from \( n \) items without regard to the order of selection.

Step 2: Use the Combination Formula

The formula to compute a combination \( C(n, k) \) is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Step 3: Calculate Factorials

Calculate the factorials needed: - \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \)- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- \( 3! = 3 \times 2 \times 1 = 6 \)

Step 4: Substitute into the Formula

Substitute \( n = 8 \) and \( k = 5 \) into the combination formula: \[ C(8, 5) = \frac{8!}{5! \, \times \, 3!} = \frac{40320}{120 \, \times \, 6} \]

Step 5: Perform the Division

Calculate \( 120 \times 6 = 720 \) and then divide \( 40320 \) by \( 720 \): \[ C(8, 5) = \frac{40320}{720} = 56 \]

Step 6: Interpret the Result

The value \( 56 \) means there are 56 different ways to choose 5 items from a set of 8 items without regard to order.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in permutations and combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a certain number. For example, the factorial of 5, written as \(5!\), is calculated as: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials grow very quickly with larger numbers, which is why they are useful in counting problems. When you want to calculate how many ways something can be arranged, factorials often provide the answer.

• 0! = 1: By convention, the factorial of zero is 1. This is useful in combinations and permutations.
• Recursive Nature: Notice that each factorial builds on the previous one: \(n! = n \times (n-1)!\).

The ability to calculate factorials is crucial for understanding more complex combinatorial formulas like the combination formula.

Combination formula

The combination formula is pivotal when figuring out how many ways you can select a group of items from a larger set, where order does not matter. In our original exercise, we are asked to find \(C(8, 5)\).
This is where the combination formula comes in. Here's how it's structured:

• Formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]
• Purpose: The formula calculates how many distinct ways you can select \(k\) items from a set of \(n\) items without considering the order.
• Components: It involves factorials of the total number, the chosen number, and their difference (\(n-k\)).

For instance, the \(C(8, 5) = 56\) calculation means there are 56 different ways to pick 5 items from 8, irrespective of order. This understanding is rooted in the combination formula.

Binomial coefficients

Binomial coefficients are the values found within Pascal's Triangle and have a variety of applications in algebra and probability. They are represented by \(C(n, k)\) or \(\binom{n}{k}\).
Here's why they're important:

• Links to Factorials: Derived using factorials, these coefficients use the formula \(C(n, k) = \frac{n!}{k!(n-k)!}\).
• In Algebra: Used in the expansion of binomials according to the Binomial Theorem: \((a+b)^n\).
• Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\) which aligns with the idea that choosing \(k\) items and choosing \(n-k\) items results in the same number of combinations.

Thus, binomial coefficients play a significant role in combinatorics and other mathematical fields, helping people identify patterns and solve various counting problems.