Difference between revisions of "2016 AMC 10A Problems/Problem 14"
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==Solution 1== | ==Solution 1== | ||
− | The amount of twos in our sum ranges from <math>0</math> to <math>1008</math>, with differences of <math>3</math> because <math>2 \cdot 3 = lcm(2, 3)</math>. | + | The amount of twos in our sum ranges from <math>0</math> to <math>1008</math>, with differences of <math>3</math> because <math>2 \cdot 3 = \lcm(2, 3)</math>. |
The possible amount of twos is <math>\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}</math>. | The possible amount of twos is <math>\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}</math>. |
Revision as of 17:50, 13 October 2018
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 $2 \cdot 3 = \lcm(2, 3)$ (Error compiling LaTeX. Unknown error_msg).
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,
.
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.