# Midterm Practice Examples, MA453, Spring 2009, Prof. Walther

The goal of this page is to make sure everyone at least understands the practice examples given on the course website before the exam. I have listed them here and will input my thoughts as I have time to do so. Please correct or bring to my attention any mistakes you find.

Some of the problems already have comments from previous semesters. Use them critically and with caution.--Walther 11:34, 29 December 2009 (UTC)

Sample questions for the midterm:

1. How many bitstrings of length 10 have exactly four zeros?

3. How many words of length 7 contain both "a" and "b"

4. In how many ways can 6 men and 8 women be lined up such that men are not adjacent?

5. How many strings of 5 digits without repetitions contain 1 or 2 but not both?

8. How many words of length 6 are there when adjacent letters being equal is not allowed?

9. How many solutions are there to x+y+z+w = 30 if x is between 5 and 10 and y is at least 6?

10. Find the probability of getting 3 of a kind but nothing better.

11. What is the probability that 2 people play poker against each other and both get 4 of a kind?

15. Find the recurrence for bitstrings that contain 0.

17. How many permutations of the English alphabet do contain "fish" but not "rat"?

18. Prove by induction that 3*11^n + 2*6^n is divisible by 5.

20. Solve a_n = 4*a_(n-1) -4*a_(n-2) with a_0=3, a_1=4.

23. Find the generating function for the Fibonacci sequence.

27. How many numbers between 1 and 10000 are not divisible by any of 5, 7, 11?