Problem 13 - Algebra and Trigonometry

Question

13–18 Evaluate the expression. $$C(8,3)$$

Step-by-Step Solution

Answer

There are 56 ways to choose 3 items from 8.

Step 1: Understanding Combinations

The expression \(C(8,3)\) represents the number of ways to choose 3 items from 8, without regard to order. This is called a combination. The formula for combinations is \(C(n, r) = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items to choose from and \(r\) is the number of items to choose.

Step 2: Applying the Formula

Substitute \(n = 8\) and \(r = 3\) into the formula: \[C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}\] This simplification takes into account the numbers to plug into the factorial formula.

Step 3: Calculating Factorials

Calculate the factorials needed: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \(3! = 3 \times 2 \times 1\), and \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 4: Simplifying the Expression

Using the simplified factorials, calculate: \[\frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56\]. We don't need to fully calculate 5! as it cancels with the factorial part in \(8!\).

Step 5: Final Result

The evaluation shows that \(C(8, 3) = 56\). Therefore, there are 56 different ways to choose 3 items out of 8.

Key Concepts

Factorials

Factorials are a way to multiply a series of descending natural numbers. The concept is denoted by an exclamation mark (!). For example, the factorial of 8, written as 8!, means you multiply 8 by each positive whole number less than it: 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. This results in a large number that's useful for counting permutations and combinations.

Factorials simplify calculations involving arrangements and selections, because they provide a way to count all possible ways to arrange a set of objects. To better understand, here are a few points:

• 0! is defined as 1 by convention, since there's exactly one way to arrange nothing.
• Factorials grow very quickly as the number increases, resulting in large values very fast.
• Commonly used in mathematical fields like algebra, calculus, and combinatorics.

Understanding how to calculate and use factorials will help you solve expressions like the given combination problem effectively.

Permutations and Combinations

While both permutations and combinations deal with arranging and selecting items, they differ in focus. Permutations consider the order of items, while combinations do not. This distinction is important for problems like the given task of evaluating C(8, 3), which focuses solely on combinations.

Combinations are calculated using the formula \(C(n, r) = \frac{n!}{r!(n-r)!}\). This formula accounts for choosing \(r\) items from \(n\) without regard to the order, which is evident in choosing 3 out of 8. Here's a simpler breakdown:

• Permutations: Useful when the order of items matters. For example, arranging 3 people in a line.
• Combinations: Useful when the order of items does not matter. Like picking 3 fruits from a basket of 8 fruits.
• The formula for permutations is \(P(n, r) = \frac{n!}{(n-r)!}\)

This concept helps in evaluating expressions like \(C(8, 3)\) by eliminating the need to consider different ordering of items, simplifying calculations.

Binomial Theorem

The Binomial Theorem is a powerful tool related to combinations that helps expand expressions of the form \((a + b)^n\). It shows how to expand these expressions into a sum involving terms of the form \(C(n, r) \, a^{(n-r)} \, b^r\), where \(C(n, r)\) is a combination that indicates the coefficients of the expanded terms.

This theorem connects directly to combinations, as it uses them to determine the coefficients during expansion. Understanding how combinations factor into the Binomial Theorem can illuminate why they are important for calculations involving multiple possibilities. Here are some key points:

• Applicable when dealing with polynomial expansions.
• Helps determine coefficients by presenting them as combinations.
• Illustrates the symmetry in polynomial expressions, with combinations making up the coefficients.

The Binomial Theorem shows the applications of combinations beyond just picking groups of items. It can be a bridge to understanding more advanced mathematical concepts.