Question

The answer is 324, show steps

2. Steve is taking a multiple-choice test consisting of five questions that each have four answers labelled A, B, C, and D. How many ways can Steve answer all five questions if he does not choose answers with the same letter for any two consecutive questions? Explain your reasoning.

Answer 1

Question 1 has 4 letters that can be chosen, so it can be answered in 4 possible ways.

Question 2 has a different letter chosen from Question 1, so 3 letters can be chosen for Question 2, so it can be answered in 3 possible ways.

Question 3 has a different letter chosen from Question 2, so 3 letters can be chosen for Question 3, so it can be answered in 3 possible ways.

Question 4 has a different letter chosen from Question 3, so 3 letters can be chosen for Question 4, so it can be answered in 3 possible ways.

Question 5 has a different letter chosen from Question 4, so 3 letters can be chosen for Question 5, so it can be answered in 3 possible ways.

Total number of ways in which 5 questions can be answered without choosing same letters for two consecutive questions = 4$\times$3$\times$3$\times$3$\times$3 = 324