Question: Test Answers

Comment on Test Answers

If we create a tree diagram with question 1 (A,B,C,D,E) and question 2 (A,B,C,D,E) and question 3 (A,B,C,D,E) and question 4 (A,B,C,D,E). I get 20. How is this approach different or incorrect?
gmat-admin's picture

I think you are drawing your tree incorrectly.
To show why I believe this, let's consider a test with only 2 questions, each having 5 answer choices.

For the first question, there are 5 answer choices. So, we have a tree with 5 branches.

Now, for EACH BRANCH that represents the first question, we have 5 more branches.

For example, if the response to the 1st question is A, then starting here, there are 5 possible branches for the response to the 2nd question. We have:
Answer A then answer A
Answer A then answer B
Answer A then answer C
Answer A then answer D
Answer A then answer E

Likewise, if the response to the 1st question is B, then from here, there are 5 possible branches for the response to the 2nd question. We have:
Answer B then answer A
Answer B then answer B
Answer B then answer C
Answer B then answer D
Answer B then answer E

And so on.
Altogether, for the 2 questions (with 5 answer choices each), we already have 25 branches.

Does that help?

Thank you for clarifying! It helped a lot! :)

I am always nervous about reading math questions wrong and has been a problem I've been noticing in my error logs. Any help would be great.

My question is couldn't this question be interpreted as how many ways could one complete the test? In that case, couldn't you answer Q4 first, Q1 second... etc...

So shouldn't the answer be 4! x 625?

I understand how you got 625, but I am just unsure of how to not misread things in the future.

gmat-admin's picture

Interesting, no one has ever suggested that interpretation. I'm not sure whether the question is ambiguous or whether it's official-GMAT-worthy.

My gut feeling is that, if two tests feature the answers Q1=B, Q2=D, Q3=A, Q4=D, then those tests are identical regardless of the order in which the questions were answered.

If we accept your interpretation as valid, then there's no end to similar interpretations. For example, what if one student answered in pen and the other in pencil? Are they different outcomes? Etc

Okay thanks. Sometimes it just takes me twice as long to do the problem again with the different interpretation when I realize none of the answer choices look right.
1Narrative's picture

Why can't we've
Stage 1; select a question (4 ways)
Stage 2: select an answer choice (5 ways).

gmat-admin's picture

Those two stages are not enough to complete the test.

Stage 1: select a question (4 ways)
Okay, I'll select question #3

Stage 2: select an answer choice (5 ways).
I'll select answer choice C.

At this point, I've done is answered ONE question. We need more steps to complete the test.

1Narrative's picture


I think picking / defining stages is one of the most difficult things to do.

Are there some basic rules? (I.e stage 1: pick question, step 2: pick answer) vs stage 1: ways to answer question 1, 2, 3, 4, 5)
gmat-admin's picture

I agree that it's tricky.

When trying to break a task into stages, ask yourself "What steps would I perform to accomplish this task?"

For example, if you need to seat 5 people in a row, then you might want to seat person #1 first, then person #2, etc.

Likewise, with completing a test, you might answer question #1 first. Then question #2, etc.

Hi Brent

Though i answered correctly, i tried solving with one different approach but i am not able to get it.
Following is my approach :

1. Scenario 1 (All answers are wrong)
I broke them into stages. I have four options to answer incorrectly a question i.e. any option other than right option. Hence, 4 ways for Q1, 4 ways for Q2... and so. Thus i get 256.

2. Scenario 2 (Only 1 answer correct)... and so on. I add up all the figures of different scenarios. But not able to get 625. Please help.
gmat-admin's picture

Your calculations for scenario 1 are correct.
I'd need to see your other calculations to see what went wrong.

That said, this approach of yours, while valid, will take a LOT of time to reach the correct answer.

I notice that there are no data sufficiency questions in this module. How frequent are counting questions on the GMAT and will there ever be any counting data sufficiency questions?
gmat-admin's picture

Great observation!

Data Sufficiency questions involving counting are VERY RARE on the GMAT. But there are a few.

Here's a video practice question:

Here's another question I created:

Here's an official question:

Hi Brent,

If it is a multiple choice question, I dont think it will be 5 ways to answer one question. Because we can have, A,B,C,D,E selected and also have AB,BC,BD,BE and ABC,ABD,ABE,...etc. So to answer one question itself will be total 40 ways. So it should be 40x40x40x40 .

gmat-admin's picture

Hi Kate,

I meant for the multiple choice questions to be similar to GMAT questions, in which we can enter only ONE response (A, B, C, D, or E) for each question. I never considered the possibility that someone could enter more than one answer for a particular question.

IF it were the case that people could submit MORE than 1 answer for a particular question, then there are 31 ways to answer a single question. Let's see why...

Outcomes with ONE selected answer:
We can enter A, B, C, D or E (5 outcomes)

Outcomes with TWO selected answers:
We can select 2 answers from 5 answers in 5C2 ways (10 ways)

Outcomes with THREE selected answers:
We can select 3 answers from 5 answers in 5C3 ways (10 ways)

Outcomes with FOUR selected answers:
We can select 4 answers from 5 answers in 5C4 ways (5 ways)

Outcomes with FIVE selected answers:
We can select 5 answers from 5 answers in 5C5 ways (1 ways)

So, the TOTAL number of ways to answer a single question = 5 + 10 + 10 + 5 + 1 = 31


Thanks Brent! This helps!!!


Hi Brent, shouldn't we also factor the order in which 4 questions can be answered i.e 4 x 3 x 2 x 1?
gmat-admin's picture

I'm happy to help.

When using the Fundamental Counting Principle, it's important to keep in mind what you're trying to accomplish with each stage.

Your solution suggests that we can complete stage 1 in 4 ways, stage 2 in 3 ways, stage 3 in 2 ways, and stage 4 in 1 way.

But, what is happening in each stage? For example, how have you defined stage 2 (which you say can be completed in 3 ways)?


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