A new flag has six vertical stripes, each stripe colored using some or all of the colors {yellow, green, blue, red}. No two adjacent stripes may have the same color. In how many distinct stripe colorings can the flag be made?

Introduction / Context:This counts sequences of length 6 over a 4-color alphabet with an adjacency restriction that prohibits identical neighbors.

Given Data / Assumptions:

• Colors available independently per stripe: 4 choices initially.
• No two consecutive stripes may have the same color.

Concept / Approach:Use multiplication with a recurrence flavor: first stripe has 4 choices; each subsequent stripe has 3 choices (anything but its immediate neighbor’s color).

Step-by-Step Solution:Stripe 1: 4 ways.Stripes 2–6: each 3 ways independently of earlier stripes beyond the immediate neighbor.Total = 4 × 3^5 = 4 × 243 = 972 = 12 × 81.

Verification / Alternative check:A simple recurrence a(n) = 3 a(n−1) with a(1) = 4 yields a(6) = 4 × 3^5.

Why Other Options Are Wrong:Other products do not equal 972 and do not match the 4·3^5 logic.

Common Pitfalls:Accidentally allowing 4 choices at each step or forbidding more than just the immediate neighbor’s color.

Final Answer:12 x 81