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# In how many ways can a teacher write an answer key for a min

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Intern
Joined: 15 Mar 2007
Posts: 14
In how many ways can a teacher write an answer key for a min  [#permalink]

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28 Oct 2007, 17:32
4
26
00:00

Difficulty:

25% (medium)

Question Stats:

74% (02:05) correct 26% (02:13) wrong based on 366 sessions

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In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

A. 88
B. 90
C. 96
D. 98
E. 102

first total possibilities no exclusions: 3t/f: 2mc
2*2*2 * 4*4 "mc" no restrictions

now compute number of exclusions:
2*1*1 *4*4 (how is this so?) wierd! How so??????

SO, 128-32= ok
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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24 Aug 2013, 00:22
23
12
aishu4 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions
followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false
questions cannot be the same?
A. 88
B. 90
C. 96
D. 98
E. 102

2 multiples-choice questions can be answered in = 4 x 4 = 16 ways
3 true-false questions can be answered in = 2x2x2 = 8 ways
But out of the 8 ways, 2 ways [(True-True-True) (False-False-False)]will contain same answers

Thus 3 true-false questions can be answered in = 2x2x2 = 6 ways
Total ways to answer the quiz = 16 x 6 = 96
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##### General Discussion
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Joined: 18 Jun 2007
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28 Oct 2007, 17:50
For TF:
3^2=9
Since 3 answers can't be the same, i subtracted 3 from the answer (post-facto, which for me means that i probably wont get this tomorrow on test day)
For MC:2^4=16
16*9=96 possibilities
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10 Aug 2011, 07:26
In how many ways can a teacher write an answer key for a mini quiz that contains 3 true false questions followed by 2 multiple choice questions with 4 answer choice each, if the correct answer to all true false questions cannot be the same?

I could come up to 2*2*2*4*4 but what next?? how can we include the constraint mentioned in the last line of the ques..pls help..
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10 Aug 2011, 09:01
Technically, the question is flawed. There is only one way in which the teacher can write an answer key provided each question has only one correct answer!

However, there are multiple ways in which a student can answer the questions.
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10 Aug 2011, 09:08
4
It's given that the correct answer for each true or false question cannot be the same. The possibility or all TRUE and all FALSE must be eliminated from the total number of possibilities.

Total number of ways in which:
a student can answer the 1st true or false question = 2

a student can answer the 2nd true or false question = 2

a student can answer the 3rd true or false question = 2

Total number of ways in which a student can answer the true or false questions = 2 x 2 x 2 = 8

Now this includes the two cases in which all answers are the same (all true and all false)

Therefore, the total number of ways in which the true and false questions such that the response for all the questions is not the same = 8 - 2 = 6

Number of ways in which the two MCQs can be answered = 4 x 4 = 16

Total number of ways in which the entire test can be answered = 16 x 6 = 96 ways...

Now... I have assumed that the student must compulsorily attempt each question

If the student has the option to skip questions, the approach is different.
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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26 Aug 2013, 10:03
fameatop wrote:
aishu4 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions
followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false
questions cannot be the same?
A. 88
B. 90
C. 96
D. 98
E. 102

2 multiples-choice questions can be answered in = 4 x 4 = 16 ways
3 true-false questions can be answered in = 2x2x2 = 8 ways
But out of the 8 ways, 2 ways [(True-True-True) (False-False-False)]will contain same answers

Thus 3 true-false questions can be answered in = 2x2x2 = 6 ways
Total ways to answer the quiz = 16 x 6 = 96

hi, you forgot to substract 2.
2x2x2 - 2 = 6 ways
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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14 Oct 2013, 22:31
fameatop wrote:
aishu4 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions
followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false
questions cannot be the same?
A. 88
B. 90
C. 96
D. 98
E. 102

2 multiples-choice questions can be answered in = 4 x 4 = 16 ways
3 true-false questions can be answered in = 2x2x2 = 8 ways
But out of the 8 ways, 2 ways [(True-True-True) (False-False-False)]will contain same answers

Thus 3 true-false questions can be answered in = 2x2x2 = 6 ways
Total ways to answer the quiz = 16 x 6 = 96

But out of the 8 ways, 2 ways [(True-True-True) (False-False-False)]will contain same answers
I don't get it! What does it mean exactly? pls help.
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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18 Nov 2013, 12:35
mww7786 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

A. 88
B. 90
C. 96
D. 98
E. 102

first total possibilities no exclusions: 3t/f: 2mc
2*2*2 * 4*4 "mc" no restrictions

now compute number of exclusions:
2*1*1 *4*4 (how is this so?) wierd! How so??????

SO, 128-32= ok

Ya, I don't get it either. Why do we need to subtract two on the True-True-True and False-False-False? Any clue?

Cheers
J
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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20 Nov 2013, 18:37
jlgdr wrote:
mww7786 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

A. 88
B. 90
C. 96
D. 98
E. 102

first total possibilities no exclusions: 3t/f: 2mc
2*2*2 * 4*4 "mc" no restrictions

now compute number of exclusions:
2*1*1 *4*4 (how is this so?) wierd! How so??????

SO, 128-32= ok

Ya, I don't get it either. Why do we need to subtract two on the True-True-True and False-False-False? Any clue?

Cheers
J

Because the question says:

In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

so you need to eliminate the t-t-t and f-f-f scenarios.
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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20 Nov 2013, 18:49
cachaval wrote:
jlgdr wrote:
mww7786 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

A. 88
B. 90
C. 96
D. 98
E. 102

first total possibilities no exclusions: 3t/f: 2mc
2*2*2 * 4*4 "mc" no restrictions

now compute number of exclusions:
2*1*1 *4*4 (how is this so?) wierd! How so??????

SO, 128-32= ok

Ya, I don't get it either. Why do we need to subtract two on the True-True-True and False-False-False? Any clue?

Cheers
J

Because the question says:

In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

so you need to eliminate the t-t-t and f-f-f scenarios.

Good catch Buddy
Many thanks
Cheers!
J
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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29 Dec 2013, 17:04
1
Actual way of solving this one is

2*2*2*4*4 all possible scenarios

2*1*1*4*4 unwanted scenarios

So it bouls down to 128 - 32 = 96

Answer is C as correctly stated above

Hope it helps!
Cheers!
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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20 Dec 2014, 10:20
Suppose I am not able to understand the condition : "if the correct answers to all true-false questions cannot be the same?"

Then 4*4 = 16 -> As there is no more condition on this part, the answer option should be an multiple of 16.
Only 96 = 16*6 is there.
So C)
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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26 Dec 2014, 03:16
Hello, I ended up in the same answer, but I think that I read the question in a totally different way... Your opinion on this would be very useful!

I used this way (trying to arrange the questions in an answer sheet):
We have 3 TF questions and 2 MC questions, which have 4 answer choices each.

The TF questions could be arranged in 3 different ways. Each one of them 1st, 2nd or 3rd
Each MC question could be arranged in 4*4= 16 ways. So, the 2 MC questions in 16*2= 32 ways.

So, 3*32= 96

Ws

Thank you.
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Posts: 58445
Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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26 Dec 2014, 09:13
pacifist85 wrote:
In how many ways can a teacher write an answer key for a mini-quiz that contains 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each, if the correct answers to all true-false questions cannot be the same?

A. 88
B. 90
C. 96
D. 98
E. 102

Hello, I ended up in the same answer, but I think that I read the question in a totally different way... Your opinion on this would be very useful!

I used this way (trying to arrange the questions in an answer sheet):
We have 3 TF questions and 2 MC questions, which have 4 answer choices each.

The TF questions could be arranged in 3 different ways. Each one of them 1st, 2nd or 3rd
Each MC question could be arranged in 4*4= 16 ways. So, the 2 MC questions in 16*2= 32 ways.

So, 3*32= 96

Ws

Thank you.

This is wrong. We have 3 true-false questions followed by 2 multiples-choice questions with 4 answer choices each:

Q1: ?

Q2: ?

Q3: ?

M1: ?
Answer: A, B, C, or D

M2: ?
Answer: A, B, C, or D

So, the answer key could be: TTF - AB, FTT - AA, ...

For 3 true-false questions there are 2*2*2 = 8 combinations possible but since we are told that the correct answers to all true-false questions cannot be the same, then we should subtract the cases TTT and FFF, so for 3 true-false questions we get 8 - 2 = 6 combinations.

For 2 multiples-choice questions with 4 answer choices each there are 4*4 = 16 combinations possible.

Thus there are total of 6*16 ways a teacher can write an answer key for the quiz.

Hope it's clear.
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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26 Dec 2014, 10:39
1
Hi pacifist85,

You did get lucky on this question, but here's how you can handle similar questions in the future.

When you're dealing with a "layered" permutation question, it's important to define what you're trying to calculate before you actually calculate it. Here, we're looking for the number of WAYS to write out the ANSWER KEY for a test that includes 3 True/False questions and 2 Four-answer Multiple Choice questions. We have an added restriction though - the 3 T/F questions CANNOT ALL have the SAME answer (so this will effect the calculation a bit since we have to eliminate some of the possibilities).

Since T/F questions have 2 possible answers, there would normally be (2)(2)(2) = 8 different sets of answers for 3 questions. You can see them in the following list:

TTT
TTF
TFT
FTT

FFF
FFT
FTF
TFF

According to the prompt, 2 of the options are NOT allowed though (TTT and FFF), so we have to remove them from consideration. That means we now have only 6 possible sets of answers for those 3 questions.

The 4-answer Multiple Choice questions are easy to deal with since there are no restrictions; there are (4)(4) = 16 possibilities there. You can also "map" them out if necessary:

BA, BB, BC, BD
CA, CB, CC, CD
DA, DB, DC, DD

The last step is to multiply the possibilities: (6)(16) = 96

Final Asnwer:

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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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24 Sep 2016, 11:01
2 choice for first 3 questions each and 4 choices for the next 2 questions. this gives us 128 total choices. here is how:
1 2 3 4 5 --->question numbers
2 2 2 4 4 = 128 --> total choices (multiplied the choices)
T F F a b ---- example key

T T T a b --- example key where first 3 questions have TTT. but there can be 16 keys in which first 3 letters would be TTT.
1 1 1 4 4 = 16

similarly for FFF:
F F F a b
1 1 1 4 4 = 16

Ways to write the key excluding the given condition = 128-16-16 = 96
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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24 Nov 2017, 16:14
I didn't quite got the 2 part of the question.

But I knew whatever happens in T/F part for multiple choice part it will be 4*4

So the answer will be a multiple of 16, and only 96 agrees.
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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26 Aug 2018, 05:40
AmrithS wrote:
It's given that the correct answer for each true or false question cannot be the same. The possibility or all TRUE and all FALSE must be eliminated from the total number of possibilities.

Total number of ways in which:
a student can answer the 1st true or false question = 2

a student can answer the 2nd true or false question = 2

a student can answer the 3rd true or false question = 2

Total number of ways in which a student can answer the true or false questions = 2 x 2 x 2 = 8

Now this includes the two cases in which all answers are the same (all true and all false)

Therefore, the total number of ways in which the true and false questions such that the response for all the questions is not the same = 8 - 2 = 6

Number of ways in which the two MCQs can be answered = 4 x 4 = 16

Total number of ways in which the entire test can be answered = 16 x 6 = 96 ways...

Now... I have assumed that the student must compulsorily attempt each question

If the student has the option to skip questions, the approach is different.

Hello .. Why are we not using 4! x 4! in this question ??
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Re: In how many ways can a teacher write an answer key for a min  [#permalink]

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04 Oct 2019, 03:52
Babar28 wrote:
AmrithS wrote:
It's given that the correct answer for each true or false question cannot be the same. The possibility or all TRUE and all FALSE must be eliminated from the total number of possibilities.

Total number of ways in which:
a student can answer the 1st true or false question = 2

a student can answer the 2nd true or false question = 2

a student can answer the 3rd true or false question = 2

Total number of ways in which a student can answer the true or false questions = 2 x 2 x 2 = 8

Now this includes the two cases in which all answers are the same (all true and all false)

Therefore, the total number of ways in which the true and false questions such that the response for all the questions is not the same = 8 - 2 = 6

Number of ways in which the two MCQs can be answered = 4 x 4 = 16

Total number of ways in which the entire test can be answered = 16 x 6 = 96 ways...

Now... I have assumed that the student must compulsorily attempt each question

If the student has the option to skip questions, the approach is different.

Hello .. Why are we not using 4! x 4! in this question ??

Why would you want to use 4! x 4!?
4! means that we can arrange 4 different things in 4! different ways. However, if you are refering to the multiply choice questions, per question we can make only one selection out of four choices say A, B, C, OR D. So, 4 possibilities for question 1 AND another 4 possibilities for question 2. Therefore, 4*4
Re: In how many ways can a teacher write an answer key for a min   [#permalink] 04 Oct 2019, 03:52
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