In how many different ways can a true-false test consisting of 9 questions be answered? There are 2 ways to answer question 1. For each of those 2 ways to answer question 1, there are 2 ways to answer question 2. That's 2*2 or 22 ways to answer questions 1 and 2. For each of those 22 ways to answer questions 1 and 2, there are 2 ways to answer question 3. That's 2*2*2 or 23 ways to answer questions 1 through 3. For each of those 23 ways to answer questions 1 through 3, there are 2 ways to answer question 4. That's 2*2*2*2 or 24 ways to answer questions 1 through 4. For each of those 24 ways to answer questions 1 through 4, there are 2 ways to answer question 5. That's 2*2*2*2*2 or 25 ways to answer questions 1 through 5. For each of those 25 ways to answer questions 1 through 5, there are 2 ways to answer question 6. That's 2*2*2*2*2*2 or 26 ways to answer questions 1 through 6. For each of those 26 ways to answer questions 1 through 6, there are 2 ways to answer question 7. That's 2*2*2*2*2*2*2 or 27 ways to answer questions 1 through 7. For each of those 27 ways to answer questions 1 through 7, there are 2 ways to answer question 8. That's 2*2*2*2*2*2*2*2 or 28 ways to answer questions 1 through 8. For each of those 28 ways to answer questions 1 through 8, there are 2 ways to answer question 9. That's 2*2*2*2*2*2*2*2*2 or 29 ways to answer questions 1 through 9. Answer: 29 = 512 ways. Edwin
In part b), for simplicity, let's reduce the problem to just two questions.
There are $2^2$ ways in which both questions may be answered true/false.
If a student does not answer question 1, there are still $2^1$ ways in which they can answer question 2, and vice versa. Thus there are $2\times 2^1$ ways in which only one question is answered.
Finally, there is just one way in which neither question is answered.
Putting this together, there are $2^2 + 2\times 2^1 + 1 = (2+1)^2 = 3^2$ ways of answering the questions.
Extending this to ten questions, there are $2^{10}$ ways to answer all ten questions, $10\times 2^9$ for answering all but one question, $45 \times 2^8$ ways of answering all but two questions, etc, giving a total of $(2+1)^{10} = 3^{10}$.
Jacqueline B. In how many ways can 10 questions on a True-False test be answered?
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In how many ways can a 10-question true-false exam be answered? (Assume that no questions are omitted)
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Each question can be answered in 2 ways: true of false. So by the Fundamental Counting Principle, a 10-question true-falce exam can be answered in: 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=210=1024 ways.
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Discrete Mathematics Basics
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