You can travel from place A to place B by 3 different buses, from B to C by 4 different buses, from C to D by 2 different buses and from D to E by 3 different buses.\nIn how many distinct ways can you travel from A to E?

Introduction / Context:
This is a basic problem on the multiplication principle of counting. We have several independent choices in sequence, and we want to know how many overall travel routes are possible from the starting point to the final destination. Each leg of the journey offers a certain number of bus options.

Given Data / Assumptions:

• From A to B: 3 possible buses.
• From B to C: 4 possible buses.
• From C to D: 2 possible buses.
• From D to E: 3 possible buses.
• Choice of bus on one leg does not restrict the choices on the other legs.
• Any combination of choices across the legs is considered a distinct way to travel.

Concept / Approach:
When there are several stages in a process and each stage offers a set number of independent options, the total number of possible outcomes is the product of the options at each stage. This is called the fundamental principle of counting or the multiplication rule. We simply multiply the number of choices for each leg of the journey to get the total number of different complete routes.

Step-by-Step Solution:
Number of choices from A to B = 3. Number of choices from B to C = 4. Number of choices from C to D = 2. Number of choices from D to E = 3. Total distinct travel routes from A to E = 3 * 4 * 2 * 3. First, 3 * 4 = 12. Then, 12 * 2 = 24. Finally, 24 * 3 = 72.

Verification / Alternative check:
We can quickly sanity check the result by thinking about a simpler version. If there were 2 choices at each of 3 legs, the total would be 2 * 2 * 2 = 8, which matches our understanding. Scaling up to 3, 4, 2 and 3 choices naturally gives a larger number, and 72 is a reasonable magnitude. There is no overlap among the different route combinations because each route is uniquely determined by the ordered list of bus choices.

Why Other Options Are Wrong:

• 36: This could come from mistakenly multiplying only three of the legs or dividing by 2 somewhere without reason.
• 64: This looks like 4^3 and arises from confusing the numbers of options with a different pattern.
• 74: This is simply an incorrect random number that does not result from any standard counting method here.

Common Pitfalls:
Some students incorrectly add instead of multiply the options, which would give 3 + 4 + 2 + 3 = 12 rather than 72. Another pitfall is thinking that the order of legs or the independence of choices somehow changes when in fact each leg of the journey is a separate choice. As long as every leg must be travelled and there are fixed independent options for each, multiplying the counts is the correct technique.

Final Answer:
The total number of distinct ways to travel from A to E is 72.