Problem 11 - Algebra and Trigonometry

Question

Evaluate each factorial expression. \(\frac{9 !}{6 !}\)

Step-by-Step Solution

Answer

504

Step 1: Understand the factorial function

A factorial of a number n, denoted as n!, is the product of all positive integers less than or equal to n. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Write the given expression with factorials expanded

The given expression is \( \frac{9!}{6!} \). Expand the factorials to see the pattern: \[ \frac{9!}{6!} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

Step 3: Simplify the common terms

Observe that \(6!\) is common in both the numerator and denominator. Cancel out these common terms: \[ \frac{9 \times 8 \times 7 \times \boxed{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{\boxed{6 \times 5 \times 4 \times 3 \times 2 \times 1}} = 9 \times 8 \times 7 \]

Step 4: Calculate the remaining product

Multiply the remaining numbers: \[ 9 \times 8 = 72 \] and then, \[ 72 \times 7 = 504 \]

Key Concepts

Factorial Function

The factorial function is an essential tool in mathematics. It helps in various areas like algebra, probability, and combinatorics. To understand a factorial function, consider the notation given as n!. This symbol stands for the product of all positive integers up to n.
For example, if n is 5, then the factorial of 5, written as 5!, is calculated as:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factoring is also used in simplifying expressions involving fractions and large numbers. Always remember that:

• Factorials grow very fast.
• They are only defined for non-negative integers.
• \(0!\) is defined as 1.

Try to practice writing factorial expressions to get used to them.

Simplifying Expressions

Simplifying expressions involving factorials makes calculations much easier. Let's take the example given in the exercise:
We need to simplify the expression \(\frac{9!}{6!}\).
First, expand both the numerator and the denominator:
\[ \frac{9!}{6!} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \].
Notice that \(6!\) is present in both the numerator and the denominator. This means they will cancel each other out.
After canceling the common terms, you're left with:
\[ 9 \times 8 \times 7 \].
This simplification greatly reduces the complexity of the problem and makes numerical calculations easier.

Numerical Multiplication

Numerical multiplication is another fundamental skill for evaluating factorial expressions. In our example, once we've simplified the expression to \(9 \times 8 \times 7\), we proceed to multiply these numbers.
First, multiply 9 by 8:
\[ 9 \times 8 = 72 \].
Next, take the result and multiply by 7:
\[ 72 \times 7 = 504 \].
This gives us the final answer:
\( \frac{9!}{6!} = 504 \).
Practicing multiplication with smaller numbers builds confidence and accuracy before moving on to larger numbers and more complex operations.