Problem 77 - Algebra and Trigonometry

Question

Explain how to find \(n !\) if \(n\) is a positive integer.

Step-by-Step Solution

Answer

To find the factorial of a positive integer \(n\) (represented as \(n!\)), start with the number \(n\) and multiply it by each positive integer less than \(n\) down to 1. For instance, \(3! = 3 \times 2 \times 1 = 6\). Similarly, in general, \(n! = n \times (n-1) \times (n-2) \times ...\times 3 \times 2 \times 1\).

Step 1: Decipher the Problem

The problem is asking to find the factorial of a positive integer \(n\). Factorial is represented with an exclamation mark (!). Factorial of a number \(n\) is the product of all positive integers less than or equal to \(n\).

Step 2: Test with a Small Number

Let's test what it would look like with a small number before using a general \(n\). For instance, let's take \(n = 3\). So, if we're trying to find \(3!\), we would do \(3 \times 2 \times 1 = 6\). The calculation starts with 3 and multiplies it by each preceding positive integer until reaching 1.

Step 3: Extend to General Case

Next, extend this concept to a general positive integer \(n\). Continuing the same pattern, the factorial of \(n\) would be calculated as follows: \(n! = n \times (n-1) \times (n-2) \times ...\times 3 \times 2 \times 1\). So, start with the number \(n\) and multiply it by each preceding positive integer until reaching 1.