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GMATPrep Basic Combinations question - Need expert help

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by voodoo_child » Wed Jun 13, 2012 4:04 am
Q :- An exam has five questions. 3 questions are multiple-choice (4 choices) and the remaining 2 questions are True/False type. Assuming that the student answers every question, how many combinations are possible?

I did this : 4^3 X 2^2 = 4^4 = 256 which is OA.

However, my colleague did this : (4*3) + (2*2) = 16. I couldn't explain him why he is wrong. As per him, for every question, there are four possibilities => 4*3. Also, for every T/F type question, there are two possibilities. Total # of combinations = 16.

Any thoughts? Please help.....

Thanks
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by aneesh.kg » Wed Jun 13, 2012 5:52 am
voodoo_child wrote:Q :- An exam has five questions. 3 questions are multiple-choice (4 choices) and the remaining 2 questions are True/False type. Assuming that the student answers every question, how many combinations are possible?

I did this : 4^3 X 2^2 = 4^4 = 256 which is OA.

However, my colleague did this : (4*3) + (2*2) = 16. I couldn't explain him why he is wrong. As per him, for every question, there are four possibilities => 4*3. Also, for every T/F type question, there are two possibilities. Total # of combinations = 16.

Any thoughts? Please help.....

Thanks
AND - Multiplication
OR - Addition

Let's rephrase the problem in the language of Permutations & Combinations.

Ask your colleague which is the better of the following two ways of rephrasing this problem:

Option 1:
Choose one out of the four choices for Q1 AND Choose one out of the four choices for Q2 AND Choose one out of the four choices for Q3 AND Choose one out of the two choices for Q4 AND Choose one out of the two choices for Q5

Option 2:
Choose one out of the four choices for Q1 AND Choose one out of the four choices for Q2 AND Choose one out of the four choices for Q3 OR Choose one out of the two choices for Q4 AND Choose one out of the two choices for Q5

Which of these two makes more sense?
Aneesh Bangia
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by gmat_and_me » Wed Jun 13, 2012 5:55 am
Your colleague should not be doing the addition. He should
do 4 * 4 * 4 * 2 * 2. If you think that there are only 3 questions
with the first having 4 choices and the last 2 having T & F as
choices, you will realize why your colleague is wrong. Just count
it!!

HTH
voodoo_child wrote:Q :- An exam has five questions. 3 questions are multiple-choice (4 choices) and the remaining 2 questions are True/False type. Assuming that the student answers every question, how many combinations are possible?

I did this : 4^3 X 2^2 = 4^4 = 256 which is OA.

However, my colleague did this : (4*3) + (2*2) = 16. I couldn't explain him why he is wrong. As per him, for every question, there are four possibilities => 4*3. Also, for every T/F type question, there are two possibilities. Total # of combinations = 16.

Any thoughts? Please help.....

Thanks
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by dimochka » Wed Jun 13, 2012 6:04 am
One interesting element of dealing with combinations (and combinatorics in general) is the meaning of mutiplication vs. Addition. Essentially, multiplication has the value of AND, whereas addition stands for OR

What your friend is essentiall doing is saying, "I can either answer any one of the multiple choice questions, with 4+4+4 different possibilities, OR I can do the true/false questions, for which I have 2+2 possibilities".
The correct way is exactly how you did it. For each possibility on the first question (4 total), there are 4 choices on the 2nd question, so just on those two there are 16 possibilities. Then for each of those, there are 4 more choices for the third multiple choice (making it 64), and similarly do the same for the two true/false.
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