Help with Counting Problem 2.1: An exam has 15 questions: 8 true/false and 7 multiple choice.

[fabaldoni]

I am working through an online course in probabilities. One of the questions is the below question. The answer I gave is the second answer, but the correct answer is the 4th answer. When you calculate the numbers they are different so I don't doubt that the 4th answer is the right one, but I have been wracking my brain trying to understand why the second answer is incorrect. I need some help understanding where I went wrong.

 

[blamocur]

What is the number of ways to pick 5 answers out of 15?
How many of those ways are unacceptable because they don't contain true/false questions?
How many are unacceptable because they do not contain multiple choice questions?

 

[Dr.Peterson]

I have been wracking my brain trying to understand why the second answer is incorrect. I need some help understanding where I went wrong.

The place to start is to write out your reasoning in choosing that answer. Without that, we can't tell where you went wrong!

But it appears that you solved a different problem: the number of ways to choose 2 t/f and 3 mc questions, in order.

The more useful question (which @blamocur has answered) is, why is the fourth answer correct?

 

[fabaldoni]

My reasoning is the following. To find all the combinations that contain at least one TF question and at least one MC question. I have 8 TF questions and 7 multiple-choice questions. So, 7x8. the last three questions can be picked from the remaining 13, hence 13x12x11. Therefore all the combinations would be 8x7x13x12x11.

I know that answer 4 is correct because the question comes from a Harvard online course where the correct answer is provided and that answer is the fourth.

After stepping away from this I understand where I went wrong. 13x12x11 will give me the sequences of the remaining 13 questions, not the combinations, so it counts the same collection multiple times. Q1Q2Q3 and Q2Q3Q1 would both get counted and they are the same collection.

Thanks for everyone's help.