5. If there are 3 roads from Town A to Town B and 4 roads from Town B to Town C, in how many ways can one go from Town A to Town C and back to Town A, through Town B, without passing through the same road twice?
Answer:
This problem involves counting the number of possible routes between three towns, where each route passes through a second town and does not repeat any roads. Let’s break it down into steps:
Step 1: Identify the number of roads between each pair of towns. We have 3 roads from Town A to Town B and 4 roads from Town B to Town C.
Step 2: Determine the number of possible routes from Town A to Town C through Town B without passing through the same road twice. To do this, we can use the multiplication principle, which states that if we have m ways of doing one thing and n ways of doing another, then we have m x n ways of doing both things in sequence.
So, we can start by choosing one of the 3 roads from Town A to Town B, then choose one of the 4 roads from Town B to Town C, and finally choose one of the 2 remaining roads from Town C back to Town A. This gives us a total of:
3 x 4 x 2 = 24
possible routes from Town A to Town C and back to Town A, through Town B, without passing through the same road twice.
- Therefore, there are 24 possible routes to take.
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