Problem 37 - Algebra and Trigonometry

Question

How many different three-letter codes are there if only the letters \(A, B, C, D,\) and \(E\) can be used and no letter can be used more than once?

Step-by-Step Solution

Answer

There are 60 different three-letter codes.

Step 1 - Determine the Total Letters to Use

Identify the letters available for the code: A, B, C, D, and E. That gives us 5 different letters to use.

Step 2 - Determine the Number of Choices for the First Letter

Since no letter can be repeated, for the first position in the code, we have 5 possible choices (A, B, C, D, E).

Step 3 - Determine the Number of Choices for the Second Letter

After choosing the first letter, only 4 letters remain for the second position.

Step 4 - Determine the Number of Choices for the Third Letter

After choosing the first and second letters, only 3 letters remain for the third position.

Step 5 - Calculate the Total Number of Codes

Multiply the number of choices for each of the three positions to find the total number of different codes: \[ 5 \times 4 \times 3 = 60 \]

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in permutations and combinatorics. The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. It is defined as: