MCS  344: Topics in Math: Combinatorics
Fall 2009

MCS-344 homepage Reading and class schedule Class problems
Prof. Barbara Kaiser Proof checklist Sample Test Problems



Sample Test Problems



  1. How many integers between 1 and 10000 are divisble by at least one of 2,3, and 5?  By at least one of 2,3,5,and 7?  Deduce that there aree at most 2288 primes less than 10000.
  2. Find a formula for the sum of the first n squares.  Prove it by induction.
  3. Suppose you have a standard deck of 52 cards.  How many 5-card hands are there that have at least one face card?  How many 5-card hands are there that have only red cards?  How many 5-card hands have at least one card in each suit?
  4. Bogart, problem 63
  5. Bogart, chapter 1 supplementary problems: 1,3,7,10,11,12
  6. Show that any set of 7 distinct integers contains a pair whose sum or difference is a multiple of 10.
  7. Show that if there are N people at a party then at least two of them know the same number of people, for N >= 2.
  8. A professor tells 3 jokes in her caclulus class every semester.  How large of a repertoire of jokes must she have in order to guarantee she never repeats the same set of three jokes over a period of 12 semesters?
  9. A bridge in a connected graph is an edge whose removal makes the graph disconnected.  Can the two vertices on a bridge have even degree?  Why or why not?
  10. Bogart, chapter 2 supplementary problems: 3, 4, 5,9
  11. Suppose you have a list of n prime numbers and a very large integer, k.  How many integers are there that are less than k and not  divisible by any of the n primes?
  12. How many social security numbers (9-digit sequences) are there such that each of the digits 1,3, and 7 appear at least once?
  13. Suppose you have n balls, numbered 1 through n, in an urn, and you remove them suequentially.  A rencontre occurs if the mth ball removed is numbered m, for some m with 1 <= m <= n.  Find the probability of getting exactly 1 rencontre and the probablility of getting at least one rencontre.  What is the probability of getting k rencontres?
  14. Use a generating function to find the number of ways to select n hotdogs given that there are 5 different varieties.
  15. Suppose we have 6 distinct dice (the normal 6-sided variety).   Find the generating function to model the number of ways we can roll these dice and have their faces sum to n, assuming that the the first three dice must be odd and the other three must be odd.
  16. How many different committees of 40 Senators can be formed (from 100) if tthe two Senators from the same state are considered identical?




Last modified: 12/11/9