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

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28 Oct 2007, 16:32

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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?

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

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..

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.
_________________

"Wherever you go, go with all your heart" - Confucius

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.
_________________

"Wherever you go, go with all your heart" - Confucius

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

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23 Aug 2013, 23:22

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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 Answer 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 Aug 2013, 09:03

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 Answer C

hi, you forgot to substract 2. 2x2x2 - 2 = 6 ways but kudos for your explanation.

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

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14 Oct 2013, 21:31

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 Answer C

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.

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

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18 Nov 2013, 11:35

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?

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

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20 Nov 2013, 17:37

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?

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.

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

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20 Nov 2013, 17:49

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?

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.

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

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20 Dec 2014, 09:20

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)
_________________

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

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26 Dec 2014, 02:16

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.

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

Was this just a lucky mistake?

Thank you.

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

Q1: ? Answer: True or False

Q2: ? Answer: True or False

Q3: ? Answer: True or False

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.

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:

AA, AB, AC, AD BA, BB, BC, BD CA, CB, CC, CD DA, DB, DC, DD

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

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

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24 Sep 2016, 10:01

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
_________________