Problem 9 - Algebra for College Students

Question

Evaluate each expression. $$\frac{8 !}{5 ! 3 !}$$

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1 - Understand the Notation

In this expression, the exclamation mark '!' denotes the factorial of a number. For example, \( n! \) means the product of all positive integers up to \( n \).

Step 2 - Write Out the Factorials

Identify the factorials involved in the problem. Here we have 8!, 5!, and 3!. So, let's write them out for clarity:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]\[ 3! = 3 \times 2 \times 1 \]

Step 3 - Substitute and Simplify

Substitute the factorials into the original expression and simplify each part. \[ \frac{8!}{5! \times 3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} \]Notice that the \(5!\) in the numerator and denominator cancel each other out.

Step 4 - Further Simplification

Now simplify the remaining expression: \[ \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \]Simplify by dividing:\[ \frac{336}{6} = 56 \]

Step 5 - Write the Final Answer

After simplifying, we find that the value of the expression is 56.

Key Concepts

Simplifying Expressions

When simplifying expressions, breaking down each component is essential. This approach helps us understand the problem better and identify potential ways to make calculations easier. In our example, we started with the expression \(\frac{8!}{5! \times 3!}\). By writing out each factorial, we can see opportunities to cancel terms and reduce the complexity.

Just like when dealing with fractions, simplifying works best when we meticulously cancel out common terms. In our example, we simplified \(\frac{8!}{5! \times 3!}\) by canceling out terms step-by-step to make further calculations manageable.

• Always start by expressing each term in its expanded form.
• Look out for common factors that can be canceled out.
• Simplify the resulting expression to the lowest terms for a final solution.

Simplifying expressions becomes more intuitive with practice, especially when managing larger numbers or more complex terms.

Factorials

Factorials are fundamental in many areas of mathematics, especially in combinatorics and calculus. A factorial, represented by \(n!\), is the product of all positive integers from 1 to \(n\). For instance, \(4!\) is equal to \(4 \times 3 \times 2 \times 1 \), which equals 24.

• Basic definition: \(n! = n \times (n-1) \times (n-2) \times ... \times 1\).
• sp\Important properties: \(0! = 1\) (by definition).

In the provided exercise, understanding factorials involved breaking down 8!, 5!, and 3! into their products. This detailed expansion allowed us to identify and cancel out common terms, which simplifies the computation of our combinatorial expression.

Factorials grow very quickly. For example, \(10!\) is already 3,628,800. Mastering these basics of factorial notation will help you succeed in other mathematical areas.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. It helps us understand how to count objects efficiently, manage arrangements, and determine the number of ways to choose items from a set. The exercise we tackled is a perfect example of using combinatorial principles.

The expression \(\frac{8!}{5! \times 3!}\) is a combinatorial formula used to find the number of ways to choose 3 items from 8 (order doesn't matter). This is known as a binomial coefficient, typically written as \({8 \brace 3}\).

• Combinations: Order does not matter, calculated as \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \).
• Permutations: Order matters, calculated as \( P(n, k) = \frac{n!}{(n-k)!} \).

Understanding these concepts is useful in statistics, probability, and various fields that require rigorous counting methods. By practicing these problems, you sharpen your ability to approach and solve complex problems systematically.