2013 AMC 12B Problems/Problem 12
Problem 12
Cities ,
,
,
, and
are connected by roads
,
,
,
,
,
, and
. How many different routes are there from
to
that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
Solution
Note that cities and
can be removed when counting paths because if a path goes in to
or
, there is only one possible path to take out of cities
or
.
So the diagram is as follows:
Now we proceed with casework. Remember that there are two ways to travel from to
,
to
,
to
and
to
.:
Case 1 : From
, if the path returns to
, then the next path must go to
. There are
possibilities of the path
. If the path goes to
from
, then the path must continue with either
or
. There are
possibilities. So, this case gives
different possibilities.
Case 2 : The path must continue with
. There are
possibilities for this case.
Putting the two cases together gives
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |