Problem 63 - College Algebra

Question

Solve by the method of your choice. How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F,\) and \(G\) if no repetition of letters is allowed?

Step-by-Step Solution

Answer

The total number of different four-letter passwords that can be formed from the given letters is \(7 * 6 * 5 * 4 = 840\).

Step 1: Understanding Permutations

Permutations refers to the arrangement of items in a specific order. In this context, it's about creating a password by arranging the seven given letters (A, B, C, D, E, F, G) in an order to form a unique four-letter password.

Step 2: Calculating Permutations without Repetition

When repetitions are not allowed, the number of available choices reduces as each letter gets selected. For the first letter of the password, there are 7 options. Once that's chosen, there are 6 options left for the second letter. For the third letter, there are 5 options, and for the fourth and final letter, there are 4 options left. Thus, the total number of permutations is determined by multiplying the number of choices for each letter.

Step 3: Performing the Calculation

The total number of permutations without repetition for a four-letter password from the seven given letters is calculated as follows: \(7 * 6 * 5 * 4\).