In a Question paper there are 4 multiple choice questions

Question

In a Question paper there are 4 multiple choice questions. Each question has 5 choices with only one choice as the correct answer. What is the total number of ways in which a candidate will not get all the four answers correct?

A. 19
B. 85
C. 120
D. 624
E. 1024

A. 19
B. 85
C. 120
D. 624
E. 1024

Bunuel · Accepted Answer

A candidate can answer the test in 5*5*5*5=5^4 number of ways (each question has 5 choices and we have total of 4 questions). Now, out of these cases there will be only one case when the candidate answered all the four questions correct. Therefore the total number of ways in which a candidate will NOT get all the four answers correct is 5^4-1=624.

Answer: D.

Hope it's clear.

avaneeshvyas · Answer

I tried this solving by the following method:
since there are 4 ways of getting the question wrong, hence the total no. of combinations required would be 
4*4*4*4*4(5 times) = 1024..... which is wrong...so please explain the flaw in my understanding

avaneeshvyas · Answer

Very stupid indeed on my part...need to sleep i guess...... :oops:

AkshdeepS · Answer

Bunuel,

I could not understand how did you calculate that there will be only one case when the candidate answered all the four questions correct.

Bunuel · Answer

There is only one correct answer to each question. So, there is 1*1*1*1=1 way to answer all 4 questions correctly.

jwalantudesai · Answer

IMO:

4 questions with each having 5 options.
There are total 5 ^4 patterns in which a student can respond.

Hence total ways of responding-5^4= 625
Now, there is 1 way which contains all 4 answers incorrect by a student.

Hence we need to subtract that-
Hence Answer = 5^4 -1= 625-1 = 624

IMO option D is answer

JeffTargetTestPrep · Answer

We can use the following equation:

Total number of ways to answer all questions - number of ways to get all questions correct = the total number of ways in which a candidate will not get all four answers correct

5 x 5 x 5 x 5 - 1 x 1 x 1 x 1 = 625 - 1 = 624

Answer: D

BrentGMATPrepNow · Answer

Let's first determine the TOTAL number of ways the test can be completed. 
The 1st question can be answered in  5  different ways (A, B, C, D, or E). 
The 2nd question can be answered in  5  different ways (A, B, C, D, or E). 
The 3rd question can be answered in  5  different ways (A, B, C, D, or E). 
The 4th question can be answered in  5  different ways (A, B, C, D, or E).

By the Fundamental Counting Principle (FCP), the total number of ways we can complete test = ( 5 )( 5 )( 5 )( 5 ) =   625   ways

So, there   625   possible outcomes
Among those   625   possible outcomes,  ONLY 1 outcome  is such that all four questions ARE answered correctly. 
This means, in the remaining  624  outcomes, the four questions are NOT all answered correctly.

Answer: D

Cheers, 
Brent

AxiomML · Answer

Is there another, albeit less elegant, way to solve this problem?

You cannot answer all 4 questions correctly, so I was trying to add the combinations of 3 correct, 2 correct, 1 correct and none correct and adding them together.

For example, for 3 correct, you have 3*(5C1) (3 correct answers) * 1*(5C4) (for the incorrect answer).

Can't seem to get the correct answer using this method though. Thoughts?

jnootenboom · Answer

I got this question wrong because I attempted to use factorials (i.e., 5! * 4!).

Can someone please explain when it is appropriate to use factorials, and when we should simply multiply the number of options together repeatedly (like in this question)?

Thanks!

mrabadia · Answer

Hi Bunuel,

Why shouldn't we include the internal arrangement of those 4 questions? I see that in your method, the order of the questions doesn't matter. How do we know when to consider the order and when not to?

Thanks in advance.

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In a Question paper there are 4 multiple choice questions

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