Difference between revisions of "2019 AMC 10C Problems/Problem 23"
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<math> 102 \pmod {1000}</math> | <math> 102 \pmod {1000}</math> | ||
Thus, the answer is <math>1</math>. | Thus, the answer is <math>1</math>. | ||
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| + | ==Video Solution== | ||
| + | https://youtu.be/U3v0LeuA9SA | ||
| + | |||
| + | ~IceMatrix | ||
Latest revision as of 18:09, 27 January 2020
Problem
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?
Solution
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 
.
Video Solution
~IceMatrix
