Counting problem - Multiple choice test

In summary, a quiz with 4 questions and 3 choices for each answer can be completed in 81 different ways if all answers are guessed. To guarantee at least 3 identical answer sheets, 163 students must take the test. This can be achieved by having 81 pairs of students who have filled out the test in all possible ways, and adding one more student for a total of 163.
  • #1
yakin
42
0
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.
 
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  • #2
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.
 
  • #3
MarkFL said:
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 :)
 

FAQ: Counting problem - Multiple choice test

What is the counting problem in a multiple choice test?

The counting problem in a multiple choice test refers to the question of how many different ways a set of choices can be combined to make the correct answer for a given question.

How do you calculate the number of possible combinations in a multiple choice test?

The number of possible combinations in a multiple choice test can be calculated by multiplying the number of choices in each question. For example, if a question has 4 choices, another question has 3 choices, and a third question has 5 choices, the total number of combinations would be 4 x 3 x 5 = 60.

Why is the counting problem important in a multiple choice test?

The counting problem is important in a multiple choice test because it helps educators and test makers ensure that the test is fair and balanced. By understanding the different combinations of choices, they can create questions that accurately assess a student's knowledge and understanding.

What is the difference between a permutation and a combination in the counting problem?

A permutation refers to the number of ways to arrange a set of choices in a specific order, while a combination refers to the number of ways to select a subset of choices without regard to order. In a multiple choice test, permutations are used to calculate the number of possible correct answers, while combinations are used to calculate the number of possible incorrect answers.

How can the counting problem be used to prevent cheating on a multiple choice test?

The counting problem can be used to prevent cheating on a multiple choice test by creating a large pool of questions with various combinations of choices. This reduces the likelihood of two students having the same exact test, making it more difficult for them to cheat off of each other.

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