There are 4 different rods between town A and Town B, 3 different roads between B and C, and 2 different roads between A and C.
Question:
How many different routes are there from A to C and back, which do not use any road twice?
Here routes and road are two different concepts?
Routes means total path from the A to C and back.
Road between two Town
Correct?
Total routes from the A to B = 4*3 + 2 = 14
Total routes from the A to B and back = 14 * 14 = 196
but can not find the answer to question
Best Answer
One straightforward though slightly tedious way is to break into cases: (i) out and back through B; (ii) direct out and back; (iii) out through B, back direct; (iv) out direct, back through B.
(i) This may be the hardest one. There are $4$ ways to get to B, and for each choice there are $3$ ways to get to C. Now there are $2$ ways to get back to B, and $3$ ways to get to A. Multiply. We get $(4)(3)(2)(3)$.
(ii) This one is easy, $2$ ways.
(iii) and (iv) Same count for each. It's your turn.
Finally, add up.