2016 AMC 10A Problems/Problem 14
Problem
How many ways are there to write as the sum of twos and threes, ignoring order? (For example, and are two such ways.)
Solution 1
The amount of twos in our sum ranges from to , with differences of because .
The possible amount of twos is .
Solution 2
You can also see that you can rewrite the word problem into an equation + = . Therefore the question is just how many multiples of subtracted from 2016 will be an even number. We can see that , all the way to , and works, with being incremented by 's.Therefore, between and , the number of multiples of is .
Solution 3
We can utilize the stars-and-bars distribution technique to solve this problem. We have 2 "buckets" in which we will distribute parts of our total sum, 2016. By doing this, we know we will have "total" answers. We want every third x and second y, so we divide our previous total by 6, which will result in . We have to round down to the nearest integer, and we have to add 2 because we did not consider the 2 solutions involving x or y being 0. So, .
Video Solution
https://youtu.be/dHY8gjoYFXU?t=1058
~IceMatrix
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |
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