SOLUTION: Towns A, B, C, and D are located in such a way that there are five roads from A to B, three roads from B to C, and three roads from C to D. How many routes are there from town A to

There are 5 different roads that we can take from A to B.
That's 5 different ways to go from A to B.

For each of those 5 ways to go from A to B,
there are 3 different roads that we can take from B to C.
That's 5×3 or 15 different ways to go from A to B to C.

For each of those 5×3 or 15 ways to go from A to B to C,
there are 3 different roads that we can take from C to D.
That's 5×3×3 or 45 different ways to go from A to B to C to D.

Answer There are 45 different routes from town A to town D via towns B and C. 

This illustrates the multiplication rule, which is

"If there are M ways to do one thing, and for each of those M
ways, there are N ways to do a second thing, then there are M×N
ways to do the first thing AND the second thing.

Suppose the 5 roads from A to B are labeled p,q,r,s, and t.
Suppose the 3 roads from B to C are labeled u,v, and w.
Suppose the 3 roads from C to D are labeled x,y, and z.

Here are all 45 ways to go from A to D via B and C.

 1. From A, take road p to B, then u to C, and finally x to D.
 2. From A, take road p to B, then u to C, and finally y to D.
 3. From A, take road p to B, then u to C, and finally z to D.
 4. From A, take road p to B, then v to C, and finally x to D.
 5. From A, take road p to B, then v to C, and finally y to D.
 6. From A, take road p to B, then v to C, and finally z to D.
 7. From A, take road p to B, then w to C, and finally x to D.
 8. From A, take road p to B, then w to C, and finally y to D.
 9. From A, take road p to B, then w to C, and finally z to D.
10. From A, take road q to B, then u to C, and finally x to D.
11. From A, take road q to B, then u to C, and finally y to D.
12. From A, take road q to B, then u to C, and finally z to D.
13. From A, take road q to B, then v to C, and finally x to D.
14. From A, take road q to B, then v to C, and finally y to D.
15. From A, take road q to B, then v to C, and finally z to D.
16. From A, take road q to B, then w to C, and finally x to D.
17. From A, take road q to B, then w to C, and finally y to D.
18. From A, take road q to B, then w to C, and finally z to D.
19. From A, take road r to B, then u to C, and finally x to D.
20. From A, take road r to B, then u to C, and finally y to D.
21. From A, take road r to B, then u to C, and finally z to D.
22. From A, take road r to B, then v to C, and finally x to D.
23. From A, take road r to B, then v to C, and finally y to D.
24. From A, take road r to B, then v to C, and finally z to D.
25. From A, take road r to B, then w to C, and finally x to D.
26. From A, take road r to B, then w to C, and finally y to D.
27. From A, take road r to B, then w to C, and finally z to D.
28. From A, take road s to B, then u to C, and finally x to D.
29. From A, take road s to B, then u to C, and finally y to D.
30. From A, take road s to B, then u to C, and finally z to D.
31. From A, take road s to B, then v to C, and finally x to D.
32. From A, take road s to B, then v to C, and finally y to D.
33. From A, take road s to B, then v to C, and finally z to D.
34. From A, take road s to B, then w to C, and finally x to D.
35. From A, take road s to B, then w to C, and finally y to D.
36. From A, take road s to B, then w to C, and finally z to D.
37. From A, take road t to B, then u to C, and finally x to D.
38. From A, take road t to B, then u to C, and finally y to D.
39. From A, take road t to B, then u to C, and finally z to D.
40. From A, take road t to B, then v to C, and finally x to D.
41. From A, take road t to B, then v to C, and finally y to D.
42. From A, take road t to B, then v to C, and finally z to D.
43. From A, take road t to B, then w to C, and finally x to D.
44. From A, take road t to B, then w to C, and finally y to D.
45. From A, take road t to B, then w to C, and finally z to D.

Edwin