An examination has 5 multiple choice questions. The first three questions have 4 answer choices each and the last two questions have 6 answer choices each. How many different answer sequences are possible for the entire paper?

Introduction / Context:
This question checks basic multiplication principle in counting. For a multiple choice paper, each question can be answered in several ways. The total number of complete answer sequences is the product of the number of choices for each question. Because the first three questions and the last two questions have different numbers of answer choices, we must treat them separately and then multiply their counts.

Given Data / Assumptions:
Total number of questions = 5. Question 1, 2 and 3 each have 4 answer choices. Question 4 and 5 each have 6 answer choices. The student must choose exactly one answer for each question. All answer choices for different questions are independent.

Concept / Approach:
We use the fundamental rule of counting, also called the rule of product. If an action can be done in a ways and another independent action can be done in b ways, then the pair can be done in a * b ways. Here, each question is an independent choice. Therefore, the total number of answer sequences equals the product of the number of choices for each question: 4 * 4 * 4 * 6 * 6.

Step-by-Step Solution:
Step 1: Compute the number of ways to answer the first three questions. Step 2: Each of these three questions has 4 choices, so choices for them = 4 * 4 * 4 = 4^3. Step 3: 4^3 = 64. Step 4: Compute the number of ways to answer the last two questions. Step 5: Each of these two questions has 6 choices, so choices = 6 * 6 = 6^2. Step 6: 6^2 = 36. Step 7: Total answer sequences = ways for first three questions * ways for last two questions. Step 8: Total = 64 * 36. Step 9: Compute 64 * 36 = 64 * (30 + 6) = 64 * 30 + 64 * 6 = 1920 + 384 = 2304. Step 10: Therefore, there are 2304 possible answer sequences.

Verification / Alternative check:
We can also write the product directly as 4^3 * 6^2 and simplify. Notice that 4^3 = 64 and 6^2 = 36, so 64 * 36 is the same computation as above. Another mental check is to confirm ranges: each of the 5 questions has at least 4 choices and no more than 6, so the total number of answer sequences should lie between 4^5 = 1024 and 6^5 = 7776. The value 2304 is between these bounds, reinforcing that the result is reasonable.

Why Other Options Are Wrong:
The value 1112 does not equal any simple product of the given powers of 4 and 6 and is too small. The value 1224 is also too small and suggests incomplete multiplication. The value 2426 is not divisible nicely by 4 or 6 and does not arise from any correct product in this context. The value 1024 equals 4^5, which wrongly assumes that all 5 questions have 4 choices. Only 2304 equals 4^3 * 6^2 and is consistent with the problem statement.

Common Pitfalls:
Students sometimes mistakenly add instead of multiply the number of choices or assume all questions have the same number of options. Another frequent error is to miscalculate 64 * 36, perhaps by incorrectly distributing the product. Always remember that for independent choices, like answering different questions on an exam, we multiply the counts, not add them. Writing the expression systematically as 4^3 * 6^2 and then simplifying helps avoid arithmetic slips.

Final Answer:
The total number of different answer sequences possible is 2304.