1. Consider all bit strings of length 10.

a)How many begin with 101?

b)How many begin with 10 and end with 11?

c)How many begin with 10 or end with 11?

d)How many have exactly three 1s?

2. Suppose that a word is any string of six letters. Repeated letters are allowed. For our purposes, vowels are the letters a, e, i, o, and u.

a)How many words are there?

b)How many words begin with A or B?

c)How many words begin with a vowel and end with a vowel?

d)How many words have no vowels?

e)How many words have exactly one vowel?

3. A professor teaching a Discrete Math course gives a multiple choice quiz that has five questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professors class in order to guarantee that at least three answer sheets must be identical? (Assume that no answers are left blank.)

4. You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get

a)a pair (for example, two kings or two 5s)

b)three of a kind (for example, three 7s)

5. Use the binomial theorem to expand (x + y)⁵.