In Exercises 23–28, evaluate each factorial expression. 17!/15!

Verified step by step guidance

1. Understand that a factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For example, \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
2. Recognize that the expression \( \frac{17!}{15!} \) involves dividing two factorials.
3. Write out the factorials: \( 17! = 17 \times 16 \times 15 \times 14 \times \ldots \times 1 \) and \( 15! = 15 \times 14 \times \ldots \times 1 \).
4. Notice that \( 15! \) is a common factor in both the numerator and the denominator, so it can be canceled out.
5. After canceling \( 15! \), you are left with \( 17 \times 16 \). Multiply these two numbers to find the result.

Key Concepts

Factorial Definition

A factorial, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations, combinations, and various mathematical calculations, making them fundamental in algebra and higher mathematics.

Simplifying Factorials

When evaluating expressions involving factorials, simplification can be performed by canceling common terms. For instance, in the expression 17!/15!, we can express 17! as 17 × 16 × 15!, allowing us to cancel the 15! in the numerator and denominator, simplifying the calculation significantly.

Properties of Factorials

Factorials have specific properties that can aid in calculations, such as n! = n × (n-1)! and 0! = 1. Understanding these properties allows for easier manipulation of factorial expressions, especially when dealing with larger numbers or complex expressions in algebra.