Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ C(8,8) $$

Step-by-Step Solution

Answer

Using the combination formula, we evaluate \(C(8, 8)\) as follows: \(C(8, 8) = \frac{8!}{8! (8-8)!} = \frac{8!}{8! \times 0!} = \frac{1}{1} = 1\). Thus, \(C(8, 8) = 1\).

Step 1: Write the combination formula

First, let's write down the formula for combinations: \[C(n, k) = \frac{n!}{k! (n-k)!}\]

Step 2: Plug in n and k values

Now, we'll plug in the values for n and k: \[C(8, 8) = \frac{8!}{8! (8-8)!}\]

Step 3: Evaluate factorials

Next, we need to evaluate the factorials (8! and (8-8)!): \[C(8, 8) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) \times (1)}\]

Step 4: Simplify the expression

Now, we can simplify the expression by canceling the same factors from the numerator and the denominator: \[C(8, 8) = \frac{8!}{8! \times 0!} = \frac{1}{1} = 1\] Thus, \(C(8, 8) = 1\).

Key Concepts

Factorials

A factorial, symbolized by an exclamation point (!), is a function that multiplies a number by every number below it down to 1. For example, the factorial of 5, written as 5!, is calculated as 5 x 4 x 3 x 2 x 1, which equals 120.

When it comes to zero, it's important to remember that the factorial of 0 is always 1 by definition. This is crucial in solving combination problems, like in the given exercise \( C(8, 8) \), where \( (8-8)! \) needs to be evaluated. Even though 0! may seem counterintuitive since 0 multiplied by anything is 0, it's a defined value that simplifies many mathematical expressions.

Understanding factorials is essential when dealing with larger values of n in the combination formula, as they can greatly increase the resulting product. It's also important to understand the concept of cancelling out factorials, as seen in the exercise, where 8! in the numerator and denominator cancel each other out, leaving us with 1.

Permutations and Combinations

Permutations and combinations are two fundamental ways of arranging a number of things. While permutations focus on the arrangement of all or part of a set of objects, with attention to the order of arrangement, combinations deal with the selection of all or part of the set without considering the order.

For example, let's consider the combination formula used in the exercise \( C(n, k) = \frac{n!}{k! (n-k)!} \). This formula calculates the number of ways to choose k items from a set of n without regard to the order. Here's an easy way to remember the difference: combinations are like a fruit salad where the order of fruits doesn't matter, while permutations are like a string of decorative lights where the order does matter.

Paying attention to whether a problem requires permutations or combinations is vital. Misinterpreting this can lead to incorrect answers. In the exercise, the use of the combination formula allows for the understanding of a situation where the order doesn't play a role, which is often the case in selecting groups or teams from a larger set.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operation signs, and sometimes parentheses that represent a quantity or a relationship between quantities. In working through a problem like the exercise provided, it's critical to understand the components of the expression and how to simplify them.

The given expression \( C(8, 8) \) is a simple example of how an expression can be broken down following mathematical rules. By evaluating the expression step by step, as demonstrated in the exercise, we can simplify complex-looking equations to something as simple as the number 1. This process often involves operations such as factorials, as well as the principles of cancelling like terms and simplifying fractions.

Understanding how to work with mathematical expressions effectively is a foundational skill in mathematics. It can involve recognizing patterns, simplifying complex terms, and applying mathematical operations in the correct order, which are crucial when moving on to more advanced math topics.