Question: Solve the following problem two ways: (a) using a tree diagram and (b) using the multiplication…

Solve the following problem two ways: (a) using a tree diagram and (b) using the multiplication principle. How many 4-letter code words can be formed from the first 4 letters of the alphabet if no letter can be used more than once?

a b c d

b. Using the multiplication principle what is the number of 4-letter words that can be formed?

Solution

To solve the problem using the multiplication principle, consider the steps for forming a 4-letter code word using the first four letters of the alphabet (A, B, C, D), with no repetition allowed: 1. For the first letter, you have 4 options (A, B, C, D). 2. Once the first letter is chosen, 3 options remain for the second letter. 3. After choosing the first two letters, 2 options remain for the third letter. 4. Finally, only 1 option remains for the last letter. Using the multiplication principle, the total number of 4-letter words is: \[ 4 \times 3 \times 2 \times 1 = 24 \] Therefore, the number of 4-letter words that can be formed is 24.