1. Prove at least three of the following by mathematical induction:
2. Answer at least three of the following:
b. How many integers between 1 and 100,000 contain exactly 4 threes and 1 seven? at least 4 threes?.
c. The license plates in a certain state have three letters followed by three numbers. How many different license plates can be made? How many license plates can be made if all of them must start with either RI, BU or FE?
d. Suppose a computer generates a 5 digit number at random. What is the probability that the first digit is odd and the last digit is either 2 or 7?
e. Prove, using any method you like, that C(n,r) = n/r * C(n-1,r-1).
f. In a 3-dimensional graph, how many non-decreasing paths can be followed between the origin (0,0,0) and the point (7,3,9), if the possible directions of the moves are fixed by the x, y and z axes? g. How many ways are there to choose 5 letters from the word BRAINSTORM? How many different arrangements of 5 letters can be made if the first two letters must be vowels, and the other three consonants?
3. Consider the following statements, and write a brief essay (at least 2 pages) comparing the way we come to know that they are true. You might want to consider how you would prove or argue for each statement, how you would evaluate an article in the New York Times claiming that these statements were false, and under what circumstances you think people should change their minds about the truth or falsity of these statements.