Counting problem - Multiple choice test

[yakin]

A quiz has 4 questions with 3 choices for each answer.
If you guess every answer, in how many different ways can you complete this test?__________
How many students must take this test to guarantee that at least 3 identical answer sheets
are submitted?__________

I know how that the answer to first part is 3.3.3.3=81 and i know how to get this answer. The answer to second part is 163, however, i do not know how to get 163? Any help would be greatly appreciated.

 

[MarkFL]

I have moved this thread here as this is a much better fit, and edited the thread title to give an indication of the nature of the problem.

Yes, there are 3 ways to answer each of the 4 questions, so the number of ways to fill out the test are:

\(\displaystyle N=3^4=81\)

Now for the other part of the question, consider that you have 162 students, and in these 162, you have 81 pairs who have filled out the test in the 81 different ways possible. Then if you add another student, no matter how he/she fills out the test, it must be done in a way that 2 students have already done, so there will now be 3 identical tests.

 

[yakin]

I have moved this thread here as this is a much better fit, and edited the thread title to give an indication of the nature of the problem.

Yes, there are 3 ways to answer each of the 4 questions, so the number of ways to fill out the test are:

\(\displaystyle N=3^4=81\)

Now for the other part of the question, consider that you have 162 students, and in these 162, you have 81 pairs who have filled out the test in the 81 different ways possible. Then if you add another student, no matter how he/she fills out the test, it must be done in a way that 2 students have already done, so there will now be 3 identical tests.

Got it sir, thanks a lot :)