Problem 23 - Algebra and Trigonometry

Question

evaluate each factorial expression. $$ \frac{17 !}{15 !} $$

Step-by-Step Solution

Answer

The result of the expression \( \frac{17!}{15!} \) is 272.

Step 1: Understanding Factorial Definition

The factorial of a number is the product of all positive integers less than or equal to that number. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. It's also important to note that the factorial of zero is one (0! = 1).

Step 2: Factorial Property

A well-known property of factorials is n! = n*(n-1)!. So, this property can be used to simplify the expression \(\frac{17!}{15!}\). Apply the property, 17! can be expressed as 17 * 16 * 15!. Thus, we convert our expression into \(\frac{17 * 16 * 15!}{15!}\).

Step 3: Simplifying the Expression

After applying the factorial property in Step 2, the two 15! on the numerator and the denominator cancel each other out because any number divided by itself equals 1. So, the expression simplifies to 17 * 16.

Step 4: Calculate the Product

After simplification, we just need to calculate the product of 17 and 16 which equals 272.

Key Concepts

Factorial Definition

A factorial, denoted by an exclamation mark (!), represents the product of an integer and all the positive integers below it.

For example, the factorial of 4, written as 4!, is calculated as:

• 4 x 3 x 2 x 1 = 24

The factorial concept is fundamental in mathematics and is used in permutations, combinations, and various fields such as statistics and algebra.

The value of 0! (zero factorial) is defined as 1. This might seem counterintuitive, but this definition keeps formulas consistent and functional.

Whenever you encounter a factorial, think of it as a way to quickly multiply a sequence of descending numbers starting from a specific number down to 1.

Factorial Property

A key property of factorials helps us simplify complex expressions, making calculations easier.

This property is represented as:

• n! = n x (n-1)!

This means that the factorial of any number n can be broken down into that number multiplied by the factorial of the number right before it.

This is especially useful when simplifying expressions like \[\frac{17!}{15!}\] as it allows us to express 17! as

• 17 x 16 x 15!.

This step makes it more straightforward to spot and cancel common terms in complex expressions. Understanding this property helps with efficient calculations and deeper mathematical insight.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form, making calculations more manageable.

In problems involving factorials, like \[\frac{17!}{15!}\] the process involves identifying and cancelling terms to ease computations.

After applying the factorial property, we rewrite the expression as:

• \[\frac{17 \times 16 \times 15!}{15!}\]

The 15! on the top and bottom cancel each other, much like x/x = 1, simplifying our expression to:

• 17 x 16

Finally, calculate 17 x 16 to find 272. This straightforward approach allows you to resolve complex-looking expressions rapidly, making math engaging and fun to solve! Always look for opportunities to simplify for efficient problem-solving.