stats, find the least squares

Question

An essay test has 13 questions. Students are required to answer 11 of the 13 questions. How many different sets of questions could be? answered?

Answer 1

There are 13 questions total, and we are choosing 11, so this is 13 choose 11.

 

We can use the combination formula:

 

C(n,r) = n!/r!(n-r)!

= 13! / (11!*(13-11)!)

= 13! / (11!2!)

= (13 * 12 * 11!) / (11! * 2)

 

The 11! can be removed since it's in both the numerator and denominator:

 

= (13 * 12) / 2

=  78

 

Therefore, there are 78 different sets of questions that can be answered

Answer 2

There's a related problem that's easier to answer and is really the same problem.

How many ways can I choose 2 questions not to answer from the 13 questions?

Intuitively, these two should yield the same result -- If I pick 2 questions not to answer, that's the "same" as picking 11 questions that I will answer. If that's not intuitive, comment and I can try to explain it.

So, let's get to work answering that one!

Let's pick a random question, say, #1. There are 13 choices. I now have 12 choices left for the second question, so we might say 13 * 12 = 156 sets.

However, consider that if I pick problem 1 first, then problem 7 second, that's really the same set as picking problem 7 first, then problem 1 second. So, we should divide by 2 since we will have counted every single set twice!

 

13 * 12 / 2 = 78, as Adam states above as well.

 

This is the combination formula as well, just without explicit reference to that particular formula.

 

and of course, you can read the wikipedia article for more information! https://en.wikipedia.org/wiki/Combination