1.Consider all bit strings of length four.

1. How many begin with 01?
2. How many begin with 01andend with 10?
3. How many begin with 01orend with 10?
4. How many have exactly three 1's?

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

1. How many words are there?
2. How many words begin with a vowel?
3. How many words begin with a vowel and end with a vowel?
4. How many words have no vowels?
5. How many words have exactly one vowel?

3. A professor teaching a Discrete Math course gives a multiple choice quiz that has four questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor's class in order to guarantee that at least four answ 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

1. a pair of kings, and
2. all four kings.

5.Use the binomial theorem to expand (x + 2y)⁴.You must illustrate use of the binomial theorem.