I am stuck on how to start this exercise. Any help is welcome.
An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify that at most 1 of questions 1 and 2 may be included among the 10. How many different choices of 10 questions are there?
I have made a few attempts so far with no luck. Here is some of what I have tried.
1. $\binom{14}8 = 3003$ This is incorrect
2. $\binom{14}9 = 2002$ This is incorrect
3. $\binom{10}8 = 45 $ This is incorrect
I have tried others with similar results, so any help is appreciated. I just don't know how to get this started.
Thanks,
Tony
Divide into cases:
(i) We use neither Q1 nor Q2. Then we must choose $10$ from the remaining $12$.
(ii) We use one of Q1 or Q2. We must choose $1$ from these $2$, and $9$ from the remaining $12$.
Add the answers to (i) and (ii).
Another way: Alternately, find the total number of choices, with no restriction. Then subtract the number of bad choices, where we picked both Q1 and Q2. There are $\binom{12}{8}$ bad choices.