It says to use nth powers or nth roots

Question

On part 1 of the test, there are 10 multiple choice questions, each with c choices. On part 2 of the test there are 5 multiple choice questions, each with d choices. Assuming a person answers each question, how many answer sheets are possible?

Answer 1

If there is only one question with x choices, then there are x¹ =x  unique possible answer sheets.

 

If thee are two questions, each with x choices, then there are x² unique possible answer sheets.

 

If there are 10 questions with c choices each, that makes c¹⁰ unique possible answer sheets; 5 questions with d choices each means d⁵ unique possible unique answer sheets.

 

Combining Part 1 and Part 2, for each unique possible answer sheet of Part 1, there are d⁵ unique possible answer sheets for Part 2.  That makes the overall number of possible fully-completed unique answer sheets:

     c¹⁰d⁵

Answer 2

Hi Ashley,

 

This question is not specific enough. I can think of what they were trying to ask from their instructions to use nth powers, but my following answer assumes that they meant to assume "...a person answers each question differently". Otherwise, the answer would be the number of people who take the test.

 

In part 1, one question has c^(1) possible answers. Because there are only c choices. Two questions would then be c^2 possible answers. Continuing this patter, 10 questions would have c^10 possible answers.

 

In part 2, a similar occurrence happens. With 5 questions, there would be d^5 possibilities.

 

Combining these two possibilities, we have c^10 * d^5 possible answer sheets.

 

Hope this helps!

Answer 3

By the multiplication rule, if you have n₁, n₂,...,n_k options, then the amount of combinations without repeat is n₁xn₂x...xn_k ways to arrange the objects.

 

Therefore assuming that you have 4 multiple choice options, 10 questions in part 1, and 5 questions in part two, then you have the result

 

10x4x5 distinct ways so, 200 answer sheets are possible.