2010 AMC 12B Problems/Problem 15
Problem 15
For how many ordered triples of nonnegative integers less than
are there exactly two distinct elements in the set
, where
?
Solution
We have either ,
, or
.
For , this only occurs at
.
has only one solution, namely,
.
has five solutions between zero and nineteen,
, and
.
has nineteen integer solutions between zero and nineteen. So for
, we have
ordered triples.
For , again this only occurs at
.
has nineteen solutions,
has five solutions, and
has one solution, so again we have
ordered triples.
For , this occurs at
and
.
and
both have one solution while
has fifteen solutions.
and
both have one solution, namely,
and
, while
has twenty solutions (
only cycles as
). So we have
ordered triples.
In total we have ordered triples
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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