Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\frac{7 !}{6 !}$$

Step-by-Step Solution

Answer

The expression evaluates to 7.

Step 1: Write out the factorials

A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. Therefore, write out the expressions for each factorial: - \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)- \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Substitute into the expression

Substitute the expressions we found for \(7!\) and \(6!\) into the given expression: \[\frac{7!}{6!} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1}\]

Step 3: Simplify the fractions

Since \(6!\) is a factor of \(7!\), the \(6 \times 5 \times 4 \times 3 \times 2 \times 1\) in the numerator cancels with the same in the denominator. This simplifies the expression to:\[7\]

Step 4: Solution

After simplifying, we find that \(\frac{7!}{6!} = 7\). This step confirms that the expression evaluates to 7.

Key Concepts

Numerical Expressions

Understanding numerical expressions is crucial when dealing with mathematical problems, including those involving factorials. A numerical expression is a combination of numbers and operations (like addition, multiplication, or factorial) that represent a specific value. Factorials are a special type of numerical expression that involve multiplying a series of descending natural numbers down to one.

Simplifying Fractions

Simplifying fractions is a key mathematical skill, especially when dealing with factorial expressions. In this exercise, simplifying involves reducing the expression \(\frac{7!}{6!}\). Here's how you do it effectively:

• Recognize that \(6!\) is a complete factor of \(7!\).
• Cancel out identical terms present in the numerator and the denominator after expanding them.

This leaves you with a simplified fraction or whole number.

Mathematical Operations

Mathematical operations are actions that are carried out to manipulate numbers. They include addition, subtraction, multiplication, division, and others such as exponentiation and factorials. Factorials, represented by an exclamation mark, require multiplication of numbers in a descending order. For instance, to solve \(\frac{7!}{6!}\), you multiply the whole numbers from 7 to 1 for \(7!\), and from 6 to 1 for \(6!\). This operation is straightforward but needs careful execution to avoid errors.