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A history exam features 5 questions. 3 of the questions are multiple

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Bunuel
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A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   03 Jan 2019, 04:52
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A history exam features 5 questions. 3 of the questions are multiple-choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam?


(A) 200
(B) 290
(C) 256
(D) 278
(E) 390
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C
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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   03 Jan 2019, 06:22
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Out of 5 questions, 3 of the questions are multiple-choice with four options each and the other two questions are true or false, i.e. 2 options each

According to the Fundamental Principle of Counting, the total number of ways is equal to the product of the independent ways.
3 of the Multiple choice questions can be answered in 4x4x4 = 64 ways
2 of the True or False questions can be answered in 2x2 = 4 ways
Therefore, when Caroline selects one answer for every question, she can answer the exam in 64x4 ways = 256 ways

Hence, the Correct Answer is Option C. 256
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BrentGMATPrepNow
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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   03 Jan 2019, 08:03
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Bunuel wrote:
A history exam features 5 questions. 3 of the questions are multiple-choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam?


(A) 200
(B) 290
(C) 256
(D) 278
(E) 390


Take the task of completing the test and break it into stages.

Stage 1: Answer the 1st multiple-choice question
Since there are 4 options from which to choose, we can complete stage 1 in 4 ways

Stage 2: Answer the 2nd multiple-choice question
Since there are 4 options from which to choose, we can complete this stage in 4 ways

Stage 3: Answer the 3rd multiple-choice question
We can complete this stage in 4 ways

Stage 4: Answer the 1st true/false question
There are 2 ways we can answer this question.
So, we can complete this stage in 2 ways

Stage 5: Answer the 2nd true/false question
We can complete this stage in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus complete the test) in [color=blue](4)(4)(4)(2)(2)/color] ways (= 256 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   19 Jan 2021, 10:09
BrentGMATPrepNow wrote:
Bunuel wrote:
A history exam features 5 questions. 3 of the questions are multiple-choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam?


(A) 200
(B) 290
(C) 256
(D) 278
(E) 390


Take the task of completing the test and break it into stages.

Stage 1: Answer the 1st multiple-choice question
Since there are 4 options from which to choose, we can complete stage 1 in 4 ways

Stage 2: Answer the 2nd multiple-choice question
Since there are 4 options from which to choose, we can complete this stage in 4 ways

Stage 3: Answer the 3rd multiple-choice question
We can complete this stage in 4 ways

Stage 4: Answer the 1st true/false question
There are 2 ways we can answer this question.
So, we can complete this stage in 2 ways

Stage 5: Answer the 2nd true/false question
We can complete this stage in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus complete the test) in [color=blue](4)(4)(4)(2)(2)/color] ways (= 256 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Why haven't we taken into account the fact that the order of the questions can be changed?
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A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   19 Jan 2021, 16:23
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Hi onit ,

Even if you change the order of the questions the answer is the same. It is just a question of whether the 4's or 2's come first within 4*4*4*2*2 but that doesn't change the answer. Hope that helps!
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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   19 Jan 2021, 22:29
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Total questions: 5

Multiple choice: 3 [four options each]

Ways to answer '3' questions: 4 * 4 * 4 = 64

Other two questions: TRUE or FALSE [2 options each]

Ways to answer these '2' questions: 2 * 2 = 4

Overall ways: 64 * 4 = 256

Answer C
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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink] New post   05 May 2023, 04:49
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Re: A history exam features 5 questions. 3 of the questions are multiple [#permalink]
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