Question

Forming Codes 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 ways to write a three-letter code.

Step-by-step solution

Step 1. Given information.

Three-letter code is to be formed using the letter A, B, C, D, and E.

Repeated letters are not allowed.

Step 2. Determine the possible number of codes.

We are choosing 3 letters from 5 letters and arranging them in order. In the ordered arrangement a letter cannot be repeated.

This is an example of a permutation without repetition in which 3 objects are chosen from 5 distinct objects.

$\begin{array}{rcl}P(5,3)& =& \frac{5!}{(5-3)!}\\ & =& \frac{5!}{2!}\\ & =& \frac{5·4·3·2!}{2!}\\ & =& 60\end{array}$

Step 3. Conclusion.

There are 60 ways to write a three-letter code.