Problem 17 - Algebra 2

Question

Evaluate each expression. \(C(12,7)\)

Step-by-Step Solution

Answer

\( C(12, 7) = 792 \)

Step 1: Understand the Combination Formula

The problem asks us to evaluate the combination, which is the number of ways to choose 7 items from 12 without regard to order. The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \). Here, \( n = 12 \) and \( r = 7 \).

Step 2: Calculate Factorials

Determine the factorials needed: \( 12! \), \( 7! \), and \( (12-7)! = 5! \). Recall that \( n! \) ("n factorial") is the product of all positive integers less than or equal to \( n \).

Step 3: Calculate \( 12! \)

Calculate the value of \( 12! \), which is \( 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600 \).

Step 4: Calculate \( 7! \)

Calculate the value of \( 7! \), which is \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).

Step 5: Calculate \( 5! \)

Calculate the value of \( 5! \), which is \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 6: Substitute into the Combination Formula

Use the factorial values in the combination formula: \( C(12, 7) = \frac{12!}{7! \times 5!} = \frac{479001600}{5040 \times 120} \).

Step 7: Simplify to Find the Result

Calculate \( 5040 \times 120 = 604800 \). Then compute \( \frac{479001600}{604800} = 792 \). Thus, \( C(12, 7) = 792 \).

Key Concepts

Factorials

The concept of factorials is fundamental in understanding combinations. A factorial, denoted by an exclamation mark "!", represents the product of all positive integers up to a given number. For example, for a number \( n \), its factorial is written as \( n! \). Calculating factorials is straightforward:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
• \( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600 \)

Factorials grow extremely fast, which means large numbers quickly result in huge factorials. This property makes them very significant in calculations of permutations and combinations. Understanding factorials helps in solving problems involving arrangements and selections, where order and selection size matter.

Permutation and Combination

Permutations and combinations are different methods used to count arrangements and selections. It's essential to distinguish between these two concepts.- Permutations deal with arrangements where the order matters. For example, the arrangement of letters "ABC" is different from "BCA."- Combinations are selections where the order does not matter. Choosing 2 letters from "ABC," gives you the same pair whether it's "AB" or "BA."The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \), where:

• \( n \) is the total number of items.
• \( r \) is the number of items to choose.
• \(! \text{ (factorial symbol) } \) denotes a factorial.

The combination formula efficiently computes how many ways we can choose \( r \) items from a total of \( n \), without considering different arrangements as separate outcomes. This is particularly useful for determining possible selections in scenarios like picking teams or committees.

Binomial Coefficient

The binomial coefficient is a specific application of combinations, often seen in binomial expansion. Denoted as \( \binom{n}{r} \), it's synonymous with \( C(n, r) \).When you see \( \binom{n}{r} \), it represents the number of ways to choose \( r \) elements from a set of \( n \) elements. It's crucial in areas like probability, statistics, and even algebra, providing essential solutions for complex problems.To illustrate with an example, consider the calculation of \( C(12, 7) \):

• Calculate \( 12! \), \( 7! \), and \( 5! \).
• Substitute these factorials into the binomial coefficient formula: \( \frac{12!}{7! \times 5!} \).
• Simplify to find the result: \( 792 \).

This process finds its roots in the binomial theorem, which provides a way to expand expressions raised to a power. The binomial coefficient is influential not just in theoretical mathematics, but also in practical problem-solving.