Problem 7 - Algebra for College Students

Question

Evaluate each expression. $$\frac{5 !}{2 ! 3 !}$$

Step-by-Step Solution

Answer

10

Step 1: Understanding Factorials

Factorials are products of all positive integers up to a given number. For example, \( 5! \) is equal to \( 5 \times 4 \times 3 \times 2 \times 1 \).

Step 2: Calculate the Factorials

Compute the factorials separately: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\), \(2! = 2 \times 1 = 2\), and \(3! = 3 \times 2 \times 1 = 6\).

Step 3: Substitute the Factorials into the Expression

Substitute the computed values of the factorials into the given expression: \( \frac{5!}{2!3!} = \frac{120}{2 \times 6} \).

Step 4: Simplify the Expression

Simplify the expression by performing the multiplication and division: \( \frac{120}{2 \times 6} = \frac{120}{12} = 10 \).

Key Concepts

Factorial Definition

A factorial is a mathematical term represented by an exclamation mark (!). It means the product of all positive integers up to a specific number. For example, the factorial of 5 (written as 5!) is the product of all positive integers from 1 to 5: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials are commonly used in permutations, combinations, and other areas of discrete mathematics. They help in simplifying expressions and calculating probabilities.
Basic properties you should know:

• \(n! = n \times (n-1) \times (n-2) \times ... \times 1\)
• \(0! = 1\) by definition

Factorial Computation

Computing factorials involves multiplying a sequence of descending natural numbers. Let's revisit the exercise step-by-step.
We need to compute \(5!\), \(2!\), and \(3!\):

• First, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• Second, \(2! = 2 \times 1 = 2\)
• Third, \(3! = 3 \times 2 \times 1 = 6\)

These computations help us substitute into the original expression.

For handling larger factorials, you might use a calculator or computer program to save time.
Remember to simplify by resolving the factorials step by step.

Simplifying Expressions

After computing the necessary factorials, the next step is substituting them back into the expression. Our original problem was:
\( \frac{5!}{2!3!} \)
Let's substitute the computed factorials:
\( \frac{120}{2 \times 6} \)
Then simplify the multiplication and division:
\( \frac{120}{12} = 10 \).
Thus, the simplified value of the expression is 10.
Key points to remember:

• Always compute factorials separately before substituting them.
• Carefully follow the order of operations: parentheses first, then multiplication and division.
• Check your work to avoid calculation errors.