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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, 17:32
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In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse 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, 12832= ok
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For TF:
3^2=9
Since 3 answers can't be the same, i subtracted 3 from the answer (postfacto, 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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Probability  Answer Key!! [#permalink]
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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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Re: Probability  Answer Key!! [#permalink]
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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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Re: Probability  Answer Key!! [#permalink]
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10 Aug 2011, 09:08
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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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aishu4 wrote: In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse questions cannot be the same? A. 88 B. 90 C. 96 D. 98 E. 102
2 multipleschoice questions can be answered in = 4 x 4 = 16 ways 3 truefalse questions can be answered in = 2x2x2 = 8 ways But out of the 8 ways, 2 ways [(TrueTrueTrue) (FalseFalseFalse)]will contain same answers Thus 3 truefalse 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, 10:03
fameatop wrote: aishu4 wrote: In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse questions cannot be the same? A. 88 B. 90 C. 96 D. 98 E. 102
2 multipleschoice questions can be answered in = 4 x 4 = 16 ways 3 truefalse questions can be answered in = 2x2x2 = 8 ways But out of the 8 ways, 2 ways [(TrueTrueTrue) (FalseFalseFalse)]will contain same answers Thus 3 truefalse questions can be answered in = 2x2x2 = 6 waysTotal 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.



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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 miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse questions cannot be the same? A. 88 B. 90 C. 96 D. 98 E. 102
2 multipleschoice questions can be answered in = 4 x 4 = 16 ways 3 truefalse questions can be answered in = 2x2x2 = 8 ways But out of the 8 ways, 2 ways [(TrueTrueTrue) (FalseFalseFalse)]will contain same answers Thus 3 truefalse 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 [(TrueTrueTrue) (FalseFalseFalse)]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 miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse 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, 12832= ok Ya, I don't get it either. Why do we need to subtract two on the TrueTrueTrue and FalseFalseFalse? 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 miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse 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, 12832= ok Ya, I don't get it either. Why do we need to subtract two on the TrueTrueTrue and FalseFalseFalse? Any clue? Cheers J Because the question says: In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse questions cannot be the same?so you need to eliminate the ttt and fff 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 miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse 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, 12832= ok Ya, I don't get it either. Why do we need to subtract two on the TrueTrueTrue and FalseFalseFalse? Any clue? Cheers J Because the question says: In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse questions cannot be the same?so you need to eliminate the ttt and fff 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
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! 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 Dec 2014, 10:20
Suppose I am not able to understand the condition : "if the correct answers to all truefalse 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
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Was this just a lucky mistake?
Thank you.



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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, 09:13
pacifist85 wrote: In how many ways can a teacher write an answer key for a miniquiz that contains 3 truefalse questions followed by 2 multipleschoice questions with 4 answer choices each, if the correct answers to all truefalse 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 truefalse questions followed by 2 multipleschoice 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 truefalse questions there are 2*2*2 = 8 combinations possible but since we are told that the correct answers to all truefalse questions cannot be the same, then we should subtract the cases TTT and FFF, so for 3 truefalse questions we get 8  2 = 6 combinations. For 2 multipleschoice 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. Answer: C. 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
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 Fouranswer 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 4answer 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 Final Asnwer: GMAT assassins aren't born, they're made, Rich
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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 = 1281616 = 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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