annguyenup wrote:

Can someone explain the following solution for this question? If a student doesn't answer the question, then there should be 2^1 ways, but why is it regarded as 2*2^1 ways?

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In how many ways can the students answer a 10-question true false examination?

(a) In how many ways can the students answer a 10-question true false examination?

(b). In how many ways can the student answer the test in part (a) if it is possible to leave a question unanswered in order to avoid an extra penalty for a wrong answer

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The explanation is:

"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*2^1 ways in which they can answer question 2, and vice versa. Thus there are 2*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*2^1+1=(2+1)^2=3^2=32 ways of answering the questions.

Extending this to ten questions, there are 2^10 ways to answer all ten questions, 10*2^9 for answering all but one question, 45×2^8 ways of answering all but two questions, etc, giving a total of (2+1)^10=3^10

**Quote:**

(a) In how many ways can the students answer a 10-question true false examination?

Let's assume there are 3 questions. The answer can be true or false. you can use slot method:

2 2 2 - so there are 2 outcomes for the first question, 2 outcomes for the second, 2 outcomes for the third:2x2x2=\(2^3\)

8 outcomes overall :

true true true

true true false

true false true

true false false

false false true

false true false

false true true

false false false

**Quote:**

(b). In how many ways can the student answer the test in part (a) if it is possible to leave a question unanswered in order to avoid an extra penalty for a wrong answer

same idea here, but there are three outcomes possible. If there are 3 questions, then

3 x

3 x

3 =3^3

If there are ten questions, then \(3^10\)

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