2019 AMC 10C Problems/Problem 23
Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?
There are choices to choose the colors for first section. Without loss of generality, assume that the first group of two is Red-Blue. There are ways to choose the colors of the next section, but we would over-count by 5 if the new top and bottom are the same.Thus, there are only distinct choices for the second group of vertical blocks. Now we just have to find the hundreds digit of . Now we can do some binomial expansions to find the hundreds digit of that number: Thus, the answer is .