Problem 54 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$\frac{9 !}{7 !}$$

Step-by-Step Solution

Answer

The value of \( \frac{9!}{7!}\) is 72.

Step 1: Understand the Problem

We need to evaluate the expression \(\frac{9!}{7!}\). The exclamation point '!' denotes a factorial, which means multiplying a series of descending natural numbers.

Step 2: Write the Factorials

Factorial of a number n, represented as \(n!\), is the product of all positive integers less than or equal to n. Therefore, \(9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) and \(7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.\)

Step 3: Cancel Out Common Factors

Since \(7!\) is a common factor in both the numerator and the denominator, they can be canceled out. Thus, \( \frac{9!}{7!} = \frac{9 \cdot 8 \cdot 7!}{7!} \). Canceling \(7!\) from both parts gives us \(9 \cdot 8\).

Step 4: Simplify

Now, simply multiply the remaining factors in the numerator: \(9 \cdot 8 = 72\).

Key Concepts

Factorial

A factorial is a fundamental concept in mathematics, particularly in algebra, represented by the exclamation mark (!). It involves multiplying a series of descending natural numbers.
For any positive integer n, the factorial of n, denoted as \(n!\), is defined as the product of all positive integers less than or equal to n.
For example:
\[5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\].
Factorials grow very quickly, making them special in many mathematical functions and equations. In our exercise, the factorials of 9 and 7 were central to solving the problem.

Numerator and Denominator

In a fraction, the top number is called the numerator, and the bottom number is called the denominator. They represent parts of a whole in many mathematical problems.
For instance, in the fraction \[\frac{9!}{7!}\], 9! is the numerator and 7! is the denominator.
When evaluating such expressions, it's crucial to understand the roles of both parts. They often share common factors, which can simplify the fraction if managed correctly. In this exercise, we treated the numerators and denominators to make the problem simpler to solve.

Cancel Out Common Factors

Many mathematical problems become easier when you cancel out common factors. If a factor appears in both the numerator and the denominator, you can remove it from the equation.
For example, in \[\frac{9!}{7!} \cdot 7!}>7!\], we can cancel 7! from both the numerator and the denominator.
By doing this, the problem simplifies significantly, reducing the equation to \[9 \cdot 8\].
This technique often simplifies otherwise complicated expressions, making them more manageable and quicker to solve.

Sequential Multiplication

Sequential multiplication is the process of multiplying a series of numbers in order. This technique is frequently used, especially when dealing with factorials.
For instance, to find 9!, you multiply all positive integers up to 9: \[9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\].
Sequential multiplication is straightforward but can result in very large numbers quickly. In our exercise, after canceling out the common \(7!\), we performed sequential multiplication of 9 and 8 only, giving us \[9 \cdot 8 = 72\]. This shows how powerful and efficient sequential multiplication can be in solving factorial problems.