<p>Let’s label the roads:
3 roads from Plattsville to Ocean Heights (and back) - 1, 2, and 3
(numbers #);
4 roads from Ocean Highs to Bay Cove (and back) - A, B, C, D, and E
(letters L).</p>
<p>We are looking for all possible arrangements
(#1, L1, L2, #2) where both #'s and L’s don’t repeat (Martina does not use
the same road twice).
There are
3 choices for the first number #1,
4 choices for the first letter L1,
3 choices for the second letter L2,
2 choices for the second number #2.
According to Fundamental Counting Principle a total number of arrangements
is 3 x 4 x 3 x 2 = 72.</p>
<p>If you really badly want to “solve this by Permutations or Combinations”,
you need to use Permutations formula, because the order of terms
in an arrangement is important: AB is different from BA.</p>
<p>We’ll have to rewrite
(#1, L1, L2, #2) as
(#1, #2, L1, L2) - it has the same number of arrangements, but now similar
terms are next to each other.
(#1, #2) can be arranged in 3P2 = 6 ways
(L1, L2) can be arranged in 4P2 = 12 ways.
Total = 6 x 12 = 72.</p>