Problem 9 - Algebra and Trigonometry

Question

Evaluate each factorial expression. \(10 !\)

Step-by-Step Solution

Answer

3628800

Step 1: Understand the Factorial Notation

The notation '!' represents a factorial. A factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. Mathematically, it is defined as: \[ n! = n \times (n-1) \times (n-2) \times \times 3 \times 2 \times 1 \]

Step 2: Set Up the Expression

Write down the expression for \(10!\) based on the factorial definition: \[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 3: Multiply Step by Step

Begin multiplying the numbers step by step: \[ 10 \times 9 = 90 \]\[ 90 \times 8 = 720 \]\[ 720 \times 7 = 5040 \]\[ 5040 \times 6 = 30240 \]\[ 30240 \times 5 = 151200 \]\[ 151200 \times 4 = 604800 \]\[ 604800 \times 3 = 1814400 \]\[ 1814400 \times 2 = 3628800 \]\[ 3628800 \times 1 = 3628800 \]

Step 4: Conclusion

After performing all the multiplications, the value of \(10!\) is computed as \[ 10! = 3628800 \]

Key Concepts

Factorial Notation

When you see a number followed by an exclamation mark, like this: 3!, it means you are dealing with a 'factorial'. Factorial notation represents the product of all positive integers from the given number down to 1. For example, the factorial of 4, which is written as 4!, means you would multiply 4 by every positive integer less than it: 4 × 3 × 2 × 1. This gives us 24. Factorials are used in various mathematical fields, such as combinatorics and algebra. When evaluating factorials, start by writing down the sequence of numbers to multiply, and carefully perform each multiplication, ensuring no numbers are missed.

Multiplication

Multiplication is a basic arithmetic operation where we combine equal groups. When working with factorials, understanding how to multiply sequentially is key. Take 5!, for instance, where you must multiply 5 × 4 × 3 × 2 × 1. We start with the highest number and consistently multiply downwards. Remember: it's helpful to break down the problem into smaller steps to avoid mistakes. For instance, do 5 × 4 first, then multiply the result by 3, then by 2, and finally by 1. Always double-check your steps to ensure accuracy, especially since factorials can grow large quickly.

Integer

An integer is a whole number that can be positive, negative, or zero. For example, -3, 0, and 7 are all integers. In the context of factorials, 'integer' often refers to the non-negative whole numbers (i.e., 0, 1, 2, 3, ...). Factorials are specifically defined for these non-negative integers. The special case here is 0!, which is defined to be 1, even though we might not multiply down from anything. This definition helps in simplifying various mathematical formulas and makes understanding factorials across different contexts more consistent. So, every time you see a factorial, you’re dealing with an integer multiplied by all positive integers less than it.