Problem 43 - Algebra 2

Question

Evaluate each expression. \(\frac{6 !}{4 !}\)

Step-by-Step Solution

Answer

The result is 30.

Step 1: Understand the Factorial Function

The factorial of a number, denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 2: Calculate the Factorials

Calculate the factorials of the given numbers:- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Step 3: Divide the Factorials

Evaluate the expression by dividing the factorial of 6 by the factorial of 4:\[\frac{6!}{4!} = \frac{720}{24}\]

Step 4: Simplify the Division

Perform the division to get the simplified result:\[\frac{720}{24} = 30\]

Key Concepts

Factorial Function

The factorial function is a fundamental concept in mathematics, represented by the symbol !. It involves multiplying all positive integers up to a specific number. For example, the factorial of 3, written as 3!, is calculated as:
\[ 3! = 3 \times 2 \times 1 = 6 \]
This mathematical operation is crucial in areas like combinatorics, algebra, and calculus. Factorials grow rapidly with larger numbers, creating large products from even relatively small numbers. It's used in expressions and equations to organize and simplify data or probabilities.

Simplification of Expressions

Simplification in mathematics means reducing expressions to their simplest form. It helps in finding more manageable and interpretable results. For factorials, the concept of simplification becomes handy in avoiding unnecessary complex calculations.
For example, in the expression \( \frac{6!}{4!} \), instead of calculating \( 6! \) and \( 4! \), you can simplify directly by canceling out the factorial terms. This is because factorials have overlapping terms, and when in division, they allow for simplification.
So from:
\[ \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \]
The terms \( 4 \times 3 \times 2 \times 1 \) in the numerator and denominator cancel out, simplifying the expression to:
\[ 6 \times 5 = 30 \]

Division of Factorials

Dividing factorials is a helpful technique, especially when simplifying complex expressions or solving combinatorial problems. In problems like \( \frac{6!}{4!} \), division helps in removing common terms in the numerator and the denominator.
This process results in a term that is more straightforward to compute. When dividing factorials:

• Identify the common terms in both the numerator and the denominator.
• Cancel the common terms to simplify the expression.
• Focus on the remaining terms to complete your calculation.

For instance, in \( \frac{6!}{4!} \), by canceling \( 4! \), you simplify to \( 6 \times 5 \), making the calculation more straightforward and efficient.