2016 AMC 10A Problems/Problem 14

Problem

How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.)

$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

Solution 1

The amount of twos in our sum ranges from $0$ to $1008$, with differences of $3$ because $2 \cdot 3 = \operatorname{lcm}(2, 3)$.

The possible amount of twos is $\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}$.

Solution 2

You can also see that you can rewrite the word problem into an equation $2x$ + $3y$ = $2016$. Therefore the question is just how many multiples of $3$ subtracted from 2016 will be an even number. We can see that $x = 1008$, $y = 0$ all the way to $x = 0$, and $y = 672$ works, with $y$ being incremented by $2$'s.Therefore, between $0$ and $672$, the number of multiples of $2$ is $\boxed{\textbf{(C)}337}$.

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 $\binom{2016-1}{2-1}$ "total" answers. We want every third x and second y, so we divide our previous total by 6, which will result in $2015/6$. 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, $2015/6\implies335\implies335+2=\boxed{\textbf{(C)}337}$.

Video Solution

https://youtu.be/dHY8gjoYFXU?t=1058

~IceMatrix

See Also

2016 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All AMC 10 Problems and Solutions

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