A test requires that you answer first Part A and then either Part B or Part C. Part A consists of 4 true-false questions, Part B consists of 4 multiple-choice questions with one correct answer out of five, and Part C consists of 5 multiple-choice questions with one correct answer out of six. How many different completed answer sheets are possible?

To determine how many different completed answer sheets are possible for this test, we need to break down the problem into manageable parts and compute the number of possible answer sheets for each part of the test.

### Part A: True-False Questions
There are 4 true-false questions in Part A. For each question, there are 2 possible answers (True or False). Therefore, the number of possible ways to answer Part A is:
\[
2^4 = 16
\]

### Part B: Multiple-Choice Questions (5 options each)
There are 4 multiple-choice questions in Part B, each with 5 possible answers (since there's one correct answer out of five choices). The number of possible ways to answer Part B is:
\[
5^4 = 625
\]

### Part C: Multiple-Choice Questions (6 options each)
There are 5 multiple-choice questions in Part C, each with 6 possible answers (since there's one correct answer out of six choices). The number of possible ways to answer Part C is:
\[
6^5 = 7776
\]

### Combining Parts
The test requires answering Part A and then choosing to answer **either Part B or Part C**.

1. The number of different ways to complete the answer sheet if you choose Part B is:
\[
16 \times 625 = 10,000
\]
2. The number of different ways to complete the answer sheet if you choose Part C is:
\[
16 \times 7776 = 124,416
\]

### Total Possible Completed Answer Sheets
Since the student must complete Part A and then either Part B or Part C, the total number of different answer sheets is the sum of the two cases:
\[
10,000 + 124,416 = 134,416
\]

Thus, the total number of different completed answer sheets is \( \boxed{134,416} \).

Would you like further clarification on any step?

Here are some related questions:
1. What would happen if Part A had more than 4 true-false questions?
2. How would the answer change if Part B had 6 multiple-choice options per question instead of 5?
3. What if you had to answer both Part B and Part C instead of choosing between them?
4. What would be the impact if Part C had fewer questions?
5. How can this method be applied to a test with open-ended questions?

**Tip:** Breaking complex problems into smaller parts can make solving them more manageable.

Learn how to calculate the total number of different completed answer sheets for a test that involves 4 true-false questions and either multiple-choice questions with 5 or 6 options. This problem applies combinatorics and counting principles to arrive at a solution. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation