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,
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 while
has twenty solutions. 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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