Problem 10 - Algebra for College Students

Question

Evaluate each expression. $$\frac{9 !}{2 ! 7 !}$$

Step-by-Step Solution

Answer

The value is 36.

Step 1: Understand the Problem

Evaluate the expression \(\frac{9!}{2!7!}\). Recall that the factorial of a number \(n!\) is the product of all positive integers up to \(n\).

Step 2: Factorial Values

Calculate the factorial values. \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \(2! = 2 \times 1\), and \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 3: Simplify the Expression

Simplify the expression by canceling out the common terms in the numerator \(9!\) and denominator \(7!\). \[ \frac{9!}{2!7!} = \frac{(9 \times 8 \times 7!)}{2! \times 7!} = \frac{9 \times 8}{2!} = \frac{9 \times 8}{2 \times 1} \]

Step 4: Evaluate the Simplified Expression

Complete the calculations in the simplified expression: \[ \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \]

Key Concepts

factorials

Factorials are a fundamental concept in combinatorics and many areas of mathematics. The factorial of a number, denoted as \( n! \), is the product of all positive integers from 1 up to \( n \). For example:
\(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
Factorials grow very quickly. Thus, they are often used in problems involving permutations and combinations. It's important to be familiar with the definition and basic properties of factorials to simplify and compute expressions involving them.

simplification

Simplification in mathematics involves reducing an expression to its most compact form. For factorial expressions, this often means canceling out common terms. Consider the example from the original exercise:
\( \frac{9!}{2!7!} \)
First, express the factorial terms. Here,
\( 9! = 9 \times 8 \times 7! \)
Notice that \( 7! \) appears in both the numerator and denominator. This allows for cancellation, simplifying the expression:
\( \frac{9 \times 8 \times 7!}{2! \times 7!} = \frac{9 \times 8}{2!} \)
Another round of simplification can reduce \( 2! \) in the denominator:
\( 2! = 2 \times 1 \)
Hence, we get:
\( \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \)

cancellation

Cancellation is a helpful technique when working with factorials and other expressions. It involves removing common factors in the numerator and denominator to simplify computations. Using the same example:
\( \frac{9!}{2!7!} \)
We can cancel out \( 7! \) from both the numerator and denominator:
\( \frac{9 \times 8 \times 7!}{2! \times 7!} = \frac{9 \times 8}{2!} \)
Notice how removing \( 7! \) simplifies the problem significantly, allowing us to then focus on the remaining numbers:
\( \frac{9 \times 8}{2!} = \frac{9 \times 8}{2 \times 1} \)
Finally, we get:
\( \frac{72}{2} = 36 \)
Effective use of cancellation can make dealing with large factorials more manageable and reduce the amount of computation required.