Problem 14 - College Algebra

Question

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{4} C_{4} $$

Step-by-Step Solution

Answer

The solution to the expression \(_{4} C_{4}\) is 1.

Step 1: Identify n and r

In the expression \(_{4} C_{4}\), 4 is both the total number of items 'n' and the number of items to choose 'r'.

Step 2: Apply the formula for combinations

\(_{4} C_{4}\) is calculated as \( \frac{4!}{4!(4-4)!} \).

Step 3: Calculate the factorials

Calculate the factorial for 4, which is \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\). Then calculate the factorial for \(4-4 = 0\), and by definition, the factorial of 0 is 1.

Step 4: Substitute the calculated values into the formula

On substituting the calculated values into the formula, we get \( \frac{24}{24 \cdot 1} = 1 \).

Key Concepts

Factorials

Factorials are a fundamental part of mathematics, especially in the study of permutations and combinations. Simply put, a factorial is the product of all positive integers up to a certain number. If you see "n!", it means "n factorial". Here’s how it works:

• The factorial of 1 (1!) is simply 1.
• The factorial of 2 (2!) is 2 x 1 = 2.
• Similarly, the factorial of 3 (3!) would be 3 x 2 x 1 = 6.

Factorials are vital for calculating combinations and permutations because they provide a way to count how many different ways you can arrange or select items from a larger collection. It's also useful for understanding the number of total arrangements or sequences possible with a given set. In our example, you saw that 4 factorial (4!) was calculated as 4 x 3 x 2 x 1, resulting in 24.

Combination Formula

The combination formula is a key component in figuring out how to choose items from a larger set. The formula for combinations is represented as:\[_{n}C_{r} = \frac{n!}{r!(n-r)!}\]This might look complicated, but let's break it down:

• "n" is the total number of items you have.
• "r" is the number of items you want to choose.
• "n!" represents the factorial of n, meaning it multiplies all whole numbers from up to n.

The division by "r!" and "(n-r)!" basically adjusts for arranging the selected items and the leftover items, respectively. It ensures we count only the different groupings and not the arrangements themselves. In the solved example, the combination of choosing 4 items from a group of 4 gives us \( \frac{4!}{4! \cdot 0!} = 1 \) way, which makes sense because there's only one way to choose everything from everything.

Binomial Coefficients

Binomial coefficients often pop up in the realm of algebra and combinatorics. They're the numerical factors that arise when expanding powers of binomials. In simpler terms, they tell us how many ways we can pick and choose items in specific arrangements. Notably, these coefficients are illustrated as "Pascal's Triangle", each number being the sum of the two directly above it. The general form of a binomial coefficient is:\[_{n}C_{r} = \frac{n!}{r! \cdot (n-r)!}\]It’s the same as the combination formula, highlighting the deep connection between combinations and binomial coefficients. Why is this important? Because these coefficients are foundational in expanding expressions like \((x+y)^n\). You'll see these coefficients indicate how each term in an expansion is scaled.

• In our exercise, \(_{4}C_{4}\) equates to 1, showing that no matter the order you rearrange your group, there’s only one way to pick all of the items.
• In larger expressions, these coefficients allow us to accurately expand and solve polynomial expressions.