Combinatorics

Question

**Exercise 1.1**

*A questionnaire consists of 5 questions with 4 answer options names for each question.*

*In how many ways can this questionnaire be filled out?*

I’m not sure if the orders matter or not.

Also if the orders matter, then it must be 5 · 4 = 20?

Which is the same as P(n,r)  =  n! / (n-r)! = 5! / (5-2)! = 5! / 3! =5 · 4 = 20

But if the order dosen't matter then it must be 10 (combinations)?

If the orders matter does it mean that we count the following as two different ways; to pick an answer and then the question vs. to pick that question and then that answer? e.g. We pick question 2 and then answer a, is not the same as if we pick answer a first and then question 2?

Then in this case the order must not matter, since it wouldn’t really make sense practically that it does? So the answer is 10 right or not?

Answer 1

Let's say you have 5 boxes (representing the questions)

How many different things can you put in the first box? (Representing the answers)

4.

Next box... 4.

And so on.

4×4×4×4×4=4^5=1024 possible ways.

I usually use this reasoning for every combinatorics problem, since I don't get confused with permutation - combination ideas and such.

Here, the order matters, since putting an A in the first box is not the same as putting an A in the second box.

I can see from your reasoning that you took a strange idea, like picking the answer and then the questions, this is wrong since you only pick one answer, you don't really pick the question, the question only marks the possibilities of the answers.

Sorry if something is unclear, I am not an expert at combinatorics

Answer 2

Think about it with smaller numbers first, so you can enumerate all possibilities to get intuition

>A questionnaire consists of 3 questions with 2 answer options names for each question.*

Let's call these answer options A/B, C/D and E/F. For example, you could answer A,D,E.

>In how many ways can this questionnaire be filled out?

List all of the ways, don't calculate them. That will help you gain intuition **why** it is possible to calculate the number of ways in a certain manner