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

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.