Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$ C(9,3) $$

Step-by-Step Solution

Answer

The expression \( C(9, 3) \) evaluates to 84.

Step 1: Understand the Combination Formula

To evaluate the expression \( C(9, 3) \), we need to use the formula for combinations, which is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.

Step 2: Substitute the Values into the Combination Formula

Substitute \( n = 9 \) and \( k = 3 \) into the formula: \[ C(9, 3) = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \, 6!} \] This sets up our calculation for the number of combinations.

Step 3: Calculate Factorials

Calculate the necessary factorial expressions:- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Simplify the Expression

Since \( 9! \) includes \( 6! \) as a factor, you can simplify the expression:- \( C(9, 3) = \frac{9 \times 8 \times 7 \times 6!}{3! \times 6!} = \frac{9 \times 8 \times 7}{3!} \)- Now, cancel \( 6! \) from the numerator and the denominator.

Step 5: Solve for the Result

Now calculate directly: \[ C(9, 3) = \frac{9 \times 8 \times 7}{3!} = \frac{504}{6} = 84 \]

Step 6: Write the Final Answer

The evaluation of \( C(9, 3) \) results in 84. Therefore, there are 84 different ways to choose 3 items from a set of 9.

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics and are crucial for calculating combinations and permutations. A factorial, represented by an exclamation mark (!), is the product of all positive integers up to a given number. For instance, the factorial of 5, written as 5!, is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow rapidly with larger numbers.Understanding factorials helps in determining the total number of ways to arrange a set of items. This is because in combinations and permutations, we often need to consider all possible arrangements or selections of a set.Here are some important points to note about factorials:

• The factorial of 0 is defined to be 1 (i.e., \(0! = 1\)), which is a useful rule in combinatorial mathematics.
• Factorials are used in formulas to express the number of ways to arrange or select items.
• They are an essential part of the fundamental counting principle.

Permutations and Combinations

Permutations and combinations are methods for counting possible arrangements in a set.

Permutations

Permutations refer to arrangements where order matters. For example, the permutation of three letters A, B, and C can be ABC, ACB, BAC, BCA, CAB, and CBA. These are six different permutations of the same three letters.The general formula for permutations of \( n \) items taken \( r \) at a time is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]

Combinations

Combinations, on the other hand, are selections where order does not matter. For example, the combination of letters A, B, and C taken two at a time is AB, AC, and BC. The arrangement BA is not considered a separate combination from AB.To calculate combinations, we use the formula:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]This formula helps determine how many different ways you can choose \( k \) items from \( n \) without considering the order, which is vital for solving problems like selecting teams or distributing prizes.

Counting Principle

The Counting Principle, also known as the Rule of Product, is a fundamental rule in combinatorics used to count the number of ways two or more independent events can occur. It states that if one event can occur in \( n \) ways and another can occur independently in \( m \) ways, then the two events can occur in \( n \times m \) ways combined.This principle is extremely helpful in situations where multiple choices or combinations are involved. For example, if you are choosing an outfit with 3 options of shirts and 4 options of pants, the Counting Principle tells us there are \( 3 \times 4 = 12 \) possible combinations of shirts and pants.Some key aspects of the Counting Principle include:

• It's a basic tool for determining possible outcomes in a sequence of events.
• It applies when the outcomes of events are independent of one another.
• It ensures thorough calculations in more complex problems involving multiple steps.

This principle lowers the complexity of calculations and provides a systematic approach when solving enumerative problems in mathematics.