Problem 10 - Algebra and Trigonometry

Question

Evaluate each factorial expression. \(9 !\)

Step-by-Step Solution

Answer

9! = 362880

Step 1: Understand the Factorial Notation

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For example, the factorial of 9 (written as 9!) is the product of all positive integers from 1 to 9.

Step 2: Write the Factorial Expression

Write down the expression for 9!: \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

Step 3: Calculate the Product

Now calculate the product step-by-step: \(9 \times 8 = 72\) \(72 \times 7 = 504\) \(504 \times 6 = 3024\) \(3024 \times 5 = 15120\) \(15120 \times 4 = 60480\) \(60480 \times 3 = 181440\) \(181440 \times 2 = 362880\) \(362880 \times 1 = 362880\)

Step 4: State the Final Result

After performing all the multiplications, the result of \(9!\) is: \(9! = 362880\)

Key Concepts

factorial notation

A factorial is a fundamental mathematical concept often represented by an exclamation mark (!). It is used to express the product of all positive integers up to a certain number. For instance, if you see 9!, it means you need to multiply all numbers from 1 up to 9. Factorial notation is widely used in permutations, combinations, and various areas of mathematics and statistics. Understanding factorial notation helps to simplify complex problems into manageable multiplication sequences.

step-by-step solution

Breaking a problem down into smaller steps can make it easier to solve. Let's look at how we can solve 9! step by step. First, identify that 9! means multiplying every integer from 1 to 9. Write this down: 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. Next, perform each multiplication one at a time to reduce potential errors. For example, start with 9 × 8 to get 72. Then multiply 72 by 7 to get 504, and so on until you complete all multiplications. This process ensures accuracy and clarity.

multiplication sequence

Understanding the order of multiplication is crucial for solving factorials. Begin with the largest number and multiply it by the next smallest number. Continue this pattern until you reach 1. For example, for 9!, you start with: 9 × 8 = 72. Next, take 72 and multiply it by 7 to get 504. Continue this sequence: 504 × 6 = 3024, 3024 × 5 = 15120, 15120 × 4 = 60480, 60480 × 3 = 181440, 181440 × 2 = 362880, and finally, 362880 × 1 = 362880. Each step-by-step multiplication reinforces the systematic approach needed to correctly evaluate factorials.

evaluating factorials

Evaluating factorials involves calculating the product of a series of descending positive integers. It's essential to follow the correct sequence of multiplications to achieve an accurate result. In our example of 9!, after performing each multiplication carefully, the final product is 362880. Factorials grow rapidly with larger numbers, making them useful for calculations in probability, algebra, and calculus. Always double-check your work by verifying each multiplication step, as errors can easily compound in a lengthy sequence.