An instructor gives an exam with fourteen questions. Students are allowed to choose any ten to answer.
a. How many different choices of ten questions are there?
b. Suppose six questions require proof and eight do not.
(i) How many groups of ten questions contain four that require proof and six that do not?
(ii) How many groups of ten questions contain at least one that requires proof?
(iii) How many groups of ten questions contain at most three that require proof?
c. Suppose the exam instructions specify that at most one of questions 1 and 2 may be included among the ten. How many different choices of Len questions are there?
d. Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the ten or neither is to be included. How many different choices of ten questions are there?

(a)

To calculate:

The number of combinations of questions of 10 that can be chosen from 14 students.

Given information:

There are 14 questions in the exam and only 10 questions should be done.

Formula used:

The number of r− combinations that can be chosen from a set of n elements is given by (nr).

(nr)=n!(n−r)!×r!

Calculation:

When selecting the combination of questions, the order is not considered. Also, no repetitions in the combinations. Hence, we can substitute n=14 and r=10 into the formula

(b)

To calculate:

The number of combinations of questions of 10 that satisfies following conditions.

1. Four questions are proofs and six are other questions.
2. At least one proof in the combination.
3. At most three questions are proofs.

(c)

To calculate:

The number of different choices of ten questions including maximum one question from first and second questions.

(d)

To find:

The number of combinations of 10 questions with either both first two questions are included or both are not included.