Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\frac{9 !}{7 !}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding Factorials

A factorial, denoted by an exclamation mark (!), is the product of all positive integers from 1 up to a given number. For example, for any integer \( n \), \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \). Therefore, \( 9!\) means \( 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \). Similarly, \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 2: Simplifying the Expression

The expression \( \frac{9!}{7!} \) can be simplified by canceling out the \( 7! \) in the numerator and the denominator. So we rewrite the expression as \( \frac{9 \times 8 \times 7!}{7!} \), which simplifies to \( 9 \times 8 \) since the \( 7! \) terms cancel each other out.

Step 3: Calculating the Result

Once simplified, we calculate \( 9 \times 8 \) which is equal to 72. This gives us the final result for the expression \( \frac{9!}{7!} \).

Key Concepts

Integer Multiplication

Multiplication is a fundamental operation in mathematics. It involves finding the total number of objects in a specified number of groups. For example, if you have 3 bags and each bag contains 4 apples, you multiply 3 by 4 to find you have 12 apples in total.

Integer multiplication follows the same principle. When we multiply two whole numbers, such as 9 and 8, we are dealing with integers. These are numbers without fractional parts or decimals. The result of multiplying 9 by 8 is 72, simply adding up multiples of 9 for 8 times. This step-by-step approach makes it simpler to handle larger numbers during calculation.

Simplifying Expressions

Simplifying expressions can make calculations easier. The main principle is to reduce expressions to their simplest form to understand or solve them effectively. Take the expression \( \frac{9!}{7!} \) as an example. This expression contains factorials, which can be complex to compute directly without simplification.

When simplifying \( \frac{9!}{7!} \), you can cancel out common terms from the numerator and the denominator. The idea here is to break down \( 9! \) to include \( 7! \), i.e., \( 9 \times 8 \times 7! \). Cancelling out \( 7! \) leaves you with \( 9 \times 8 \), which is a straightforward computation. This simplification process allows for more direct calculation without handling overly large numbers.

Mathematical Notation

Mathematical notation is a system of symbols used to write formulas and equations. It helps express mathematical concepts clearly and concisely. One such notation in this context is the factorial, denoted by an exclamation mark (!). The notation \( n! \) represents a factorial, which is the product of an integer and all the integers below it, down to one.

Understanding this notation is crucial for simplifying expressions and solving equations. For example, with \( 9! \), you know that it means multiplying numbers from 9 down to 1. This concise notation allows us to convey complex ideas like the "factorial" without writing out extended multiplication sequences, aiding in both understanding and efficiency.