On an assignment, students are expected to answer at least
3 questions from a 16 question set. But they can complete
as many questions as they wish (for bonus!). How many
possible ways can they complete the assignment?
First we calculate the number of ways where they do not have
to answer at least three. Then we'll calculate the number
of ways in which exactly 0, 1, or 2 are answered. Then we'll
subtract the second number from the first.
1. We calculate the number of ways where they do not have
to answer at least three, but can answer from 0 up to 16.
There are 2 choices for question 1, answer it or don't.
There are 2 choices for question 2, answer it or don't.
There are 2 choices for question 3, answer it or don't.
...
...
There are 2 choices for question 15, answer it or don't.
There are 2 choices for question 16, answer it or don't.
So the number of ways a student can answer any number from
none at all up to all 16 is 216.
2. a. We calculate the number of ways when none are answered.
There is just one way 16C0 = 1
b. We calculate the number of ways when only 1 is answered.
There are 16 ways, 16C1 = 16
c. We calculate the number of ways when only 2 are answered.
There are 16C2 = 120 ways
Answer: 216-1-16-120 = 65399
Edwin
