Problem 19 - Algebra 2

Question

Evaluate each expression. \(C(12,4) \cdot C(8,3)\)

Step-by-Step Solution

Answer

27720

Step 1: Understand the Combination Notation

The combination notation \( C(n, k) \) refers to the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order. The formula is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \(!\) denotes factorial, the product of all positive integers up to that number.

Step 2: Calculate \( C(12, 4) \)

Using the formula for combinations, calculate \( C(12, 4) \): \[ C(12, 4) = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \, 8!} \] Now manually calculate the top (numerator) and bottom (denominator): - Top: \( 12 \times 11 \times 10 \times 9 = 11880 \) (As \(8!\) terms will cancel out)- Bottom: \( 4 \times 3 \times 2 \times 1 = 24 \)So, \( C(12, 4) = \frac{11880}{24} = 495 \).

Step 3: Calculate \( C(8, 3) \)

Next, calculate \( C(8, 3) \) using the combination formula: \[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \, 5!} \] Calculate the top and bottom: - Top: \( 8 \times 7 \times 6 = 336 \) (As \(5!\) terms will cancel out)- Bottom: \( 3 \times 2 \times 1 = 6 \)So, \( C(8, 3) = \frac{336}{6} = 56 \).

Step 4: Multiply the Results

Now multiply the results of both combinations: \( C(12, 4) \cdot C(8, 3) \).The calculation is: \[ 495 \times 56 = 27720 \].

Key Concepts

Factorial

The concept of a factorial is key to understanding combinations and permutations. A factorial is represented by an exclamation mark, e.g., \(!n\), and it's the product of all positive integers from 1 up to that number. For instance, \(5!\) means \(5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials tend to grow very large, very quickly as numbers increase.

Here's why factorials are crucial:

• Factorials help count the total number of ways to arrange a set of items.
• They're foundational in calculating combinations and permutations.

In many combination problems, calculating factorials can help you understand how large or small a potential solution set might be. Factorials also allow you to cancel terms, simplifying complex fraction calculations.

Binomial Coefficient

The binomial coefficient is a significant concept in combinatorics, frequently used in calculating combinations. It is commonly expressed as \( C(n, k) \) or \( \binom{n}{k} \), representing the number of ways to pick \( k \) elements from \( n \) elements, where the order doesn't matter. The formula to calculate the binomial coefficient is:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

This formula decomposes into three main steps:

• Finding \( n! \) as the total number of possible permutations of the set.
• Dividing by \( k! \) to account for the different permutations of the chosen elements.
• Also dividing by \( (n-k)! \) for the remaining elements not chosen.

By understanding the binomial coefficient, one can determine not just the number of combinations, but also understand the balance between selecting the number of items and arranging those selections.

Combination Formula

The combination formula enables you to determine how many ways you can choose a subset of items from a larger set, without concern for the arrangement of those items. The general combination formula is expressed as:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Where:

• \(n\) is the total number of items.
• \(k\) is the number of items to choose.
• \(!\) (factorial) helps break down complex permutations into manageable calculations.

In our exercise, we calculated \( C(12, 4) \) and \( C(8, 3) \), illustrating that the formula efficiently breaks down even seemingly large problems into straightforward calculations. Using the combination formula allows you to compute the number of groups (or sets) that can be formed, by focusing on selection rather than arrangement. This principle is used widely in fields like probability, statistics, and discrete mathematics.