2017 AIME I Problems/Problem 7
Problem 7
For nonnegative integers and
with
, let
. Let
denote the sum of all
, where
and
are nonnegative integers with
. Find the remainder when
is divided by
.
Solution
Let , and note that
. The problem thus asks for the sum
over all
such that
. Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression counts the total number of ways to select 6 dots by considering each column separately. However, this must be equal to
. Therefore, the answer is
.
-rocketscience
Solution 2 (Major Bash)
Case 1: .
Subcase 1:
Subcase 2:
Subcase 3:
Case 2:
By just switching and
in all of the above cases, we will get all of the cases such that
is true. Therefore, this case is also
Case 3:
Solution 3
Treating as
, this problem asks for:
(
*
(
*
))
But,
can be seen as the following combinatorial argument:
Choosing elements from a set of size
is the same as splitting the set into two sets of size
and choosing
elements from one,
from the other where
<=
<=
.
Thus, such a procedure is simply .
Therefore, our answer is .
- Awsomness2000
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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