In a test of 8 questions, each question has an alternative.

[RBBmba@2014]

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

[Brent@GMATPrepNow]

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

Take the task of selecting or not selecting questions and break it into stages.

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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Then you can try solving the following questions:

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MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
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- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
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- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent

[GMATGuruNY]

For a similar problem, check here:

https://www.beatthegmat.com/combination- ... tml#318126

[RBBmba@2014]

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

Take the task of selecting or not selecting questions and break it into stages.

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

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.

[Matt@VeritasPrep]

As for the question, I'd request you to clarify what does this mean - " each question has an alternative " ?

Look forward to your reply.

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).

[Brent@GMATPrepNow]

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.

As Matt points out, the question is poorly worded (what's the source?).
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

[binit]

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.

[akhil p]

It should be 3^8-1.....as the pssible ways of judging an answer is....u answer the question...or its alternative or do not answer the question...
hence 3^8....but here no. of ways to answer the question is3^8-1...(-1 because we're not taking none of the questions answered. $$$$ $$$$ $$$$