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?

Short answer

There are 60 different three-letter codes.

Step-by-step solution

- 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.

- 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).

- Determine the Number of Choices for the Second Letter

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

- Determine the Number of Choices for the Third Letter

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

- 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: