In a test of 8 questions, each question has an alternative.What is the number of ways in which one can select one or more questions ?
(A) 2^8-1
(B) 2^8
(C) 2^8*3
(D) 3^8-1
(E) 3^8
OA: D
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In a test of 8 questions, each question has an alternative.
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RBBmba@2014
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Take the task of selecting or not selecting questions and break it into stages.RBBmba@2014 wrote:In a test of 8 questions, each question has an alternative.What is the number of ways in which one can select one or more questions ?
(A) 2^8-1
(B) 2^8
(C) 2^8*3
(D) 3^8-1
(E) 3^8
Stage 1: Select or don't select question #1
We have 2 options: select the question or don't select the question.
So, we can complete stage 1 in 2 ways
Stage 2: Select or don't select question #2
We have 2 options: select the question or don't select the question.
So, we can complete stage 2 in 2 ways
Stage 3: Select or don't select question #3
We have 2 options: select the question or don't select the question.
So, we can complete stage 3 in 2 ways
.
.
.
.
Stage 8: Select or don't select question #8
We have 2 options: select the question or don't select the question.
So, we can complete stage 8 in 2 ways
By the Fundamental Counting Principle (FCP), we can complete all 8 stages (and thus deal with all 8 questions) in (2)(2)(2)(2)(2)(2)(2)(2) ways (= 2^8 ways)
So, the correct answer is B . . . . NOPE!
Included in those 2^8 ways is a single case in which NONE of the questions are selected. HOWEVER, the question tells us that we must select one or more questions. So, we should not have counted that 1 case in which NONE of the questions are selected.
So, we must subtract 1 from our answer to get [spoiler]2^8 - 1[/spoiler]
Answer: A
--------------------------
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Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Mar 11, 2015 9:31 am, edited 1 time in total.
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RBBmba@2014
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Hi Brent,Brent@GMATPrepNow wrote:Take the task of selecting or not selecting questions and break it into stages.RBBmba@2014 wrote:In a test of 8 questions, each question has an alternative.What is the number of ways in which one can select one or more questions ?
(A) 2^8-1
(B) 2^8
(C) 2^8*3
(D) 3^8-1
(E) 3^8
Stage 1: Select or don't select question #1
We have 2 options: select the question or don't select the question.
So, we can complete stage 1 in 2 ways
Stage 2: Select or don't select question #2
We have 2 options: select the question or don't select the question.
So, we can complete stage 2 in 2 ways
Stage 3: Select or don't select question #3
We have 2 options: select the question or don't select the question.
So, we can complete stage 3 in 2 ways
.
.
.
.
Stage 8: Select or don't select question #8
We have 2 options: select the question or don't select the question.
So, we can complete stage 8 in 2 ways
By the Fundamental Counting Principle (FCP), we can complete all 8 stages (and thus deal with all 8 questions) in (2)(2)(2)(2)(2)(2)(2)(2) ways (= 2^8 ways)
So, the correct answer is E . . . . NOPE!
Included in those 2^8 ways is a single case in which NONE of the questions are selected. HOWEVER, the question tells us that we must select one or more questions. So, we should not have counted that 1 case in which NONE of the questions are selected.
So, we must subtract 1 from our answer to get [spoiler]2^8 - 1[/spoiler]
Answer: D
--------------------------
Cheers,
Brent
Thanks for your reply. But I think, per your explanation the OA should be A (i.e. 2^8 - 1). Not the option D (I also marked it wrong unfortunately in my post!). Correct me please if wrong.
As for the question, I'd request you to clarify what does this mean - " each question has an alternative " ?
Look forward to your reply.
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The wording is strange, but I think it means that the test has two options for each of the eight questions (almost like a primitive CAT), but each of those options can ONLY be assigned to that specific number. In other words, we have questions {A, B, C, ..., P}, and Question 1 can be (A or B), Question 2 can be (C or D), ..., and Question 8 can be (O or P).RBBmba@2014 wrote:As for the question, I'd request you to clarify what does this mean - " each question has an alternative " ?
Look forward to your reply.
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Brent@GMATPrepNow
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As Matt points out, the question is poorly worded (what's the source?).RBBmba@2014 wrote: Hi Brent,
Thanks for your reply. But I think, per your explanation the OA should be A (i.e. 2^8 - 1). Not the option D (I also marked it wrong unfortunately in my post!). Correct me please if wrong.
As for the question, I'd request you to clarify what does this mean - " each question has an alternative " ?
Look forward to your reply.
I assumed that, for each question, we have TWO options (two alternatives): keep or don't keep.
If there are, indeed, 2 such options, then the correct answer is A.
Cheers,
Brent
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IMO the question asks to determine the no. of ways a student can choose one or more than one Qns.
So, imagine u r a student on such a test u can answer any number of Qns out of 8 Qns(each of 8 Qns has alternatives: A & B) Obviously, it's a liberal test with very little constraints.
What are your options on the Qn. no. 1?? Well, u can select 1.A or select 1.B or select nothing. So, 3 ways to select Qn no. 1.
Similarly 3 ways to select Qn no. 2 and so on up to Qn no. 8.
Together, you have 3^8 ways to select 8 questions.
But ONE way in all these is selecting NOTHING in all 8 Qns. That has to be eliminated as Qn asks one or more than one.
So, N = 3^8-1. Answer.
So, imagine u r a student on such a test u can answer any number of Qns out of 8 Qns(each of 8 Qns has alternatives: A & B) Obviously, it's a liberal test with very little constraints.
What are your options on the Qn. no. 1?? Well, u can select 1.A or select 1.B or select nothing. So, 3 ways to select Qn no. 1.
Similarly 3 ways to select Qn no. 2 and so on up to Qn no. 8.
Together, you have 3^8 ways to select 8 questions.
But ONE way in all these is selecting NOTHING in all 8 Qns. That has to be eliminated as Qn asks one or more than one.
So, N = 3^8-1. Answer.
