Fall 2010 Third Test for Tucker's AMS 301 course
1. (8 pt) Find the rook polynomial and give an expression for the number of matchings of 5 men (Rows) and 5 women (Columns) given the follow 8 conflicting pairs: (M1,W2),(M1,W5),(M2,W1),(M2,W3),(M3,W4),(M4,W3),(M4,W4),(M5,W2),(M5,W2).
2. (8 pt) How many ways are there to assign 10 different jobs to one of 3 types of computers-- Dell PC, an Apple MacIntosh or to an ApacheXX Workstation-- if at least one job must be assigned to each computer?
3. (4 pt) Find a rec. relation for an,m, the number of ways to distribute n identical hats into m different boxes with 2 or 5 or 8 hats in each box.
4. (5 pt) Find a recurrence relation for an, the number of sequences of length n formed by a's, b's and c's with the subsequence aa not allowed (no pair of consecutive ab's).
b) (5 pt) Repeat part a) but now with the requirement that you cannot have the subsequence abc.
5. (14 pt) There are eight tennis players and each week for 5 weeks, a different pair of the eight play a tennis match. How many ways are there to form the sequence of 5 matches so that every player plays at least once?


Fall 2009 Third Test for Tucker's AMS 301 course
1. Find the rook polynomial and give an expression for the number of matchings of 5 men (Rows) and 5 women (Columns) given the follow 8 conflicting pairs: (M1,W4),(M2,W2),(M3,W1),(M3,W3),(M3,W5),(M4,W3),(M5,W2),(M5,W4).
2. How many ways are there to arrange the letters in the word STATISTICS so that the three S's do not appear consecutively and the three T's are not consecutive and the two I's are not consecutive?
3. Find a rec. rel for an,k, the number of ways to select n donuts from k diff. types of donuts with 3 or 5 or 8 donuts chosen of each type.
4. a) Find a recurrence relation for an, the number of sequences of 2's, 3's and 4's whose sum is n.
b) Repeat part a) with the requirement that no 4 can be followed by a 2.
5. There are 8 Broadway musicals and they offer a special three-night package ( Friday, Saturday, Sunday nights) where one can order a single ticket that is good for three different musicals on successive nights (a SEQUENCE of three musicals). A travel agent is going to order a collection of 30 different such tickets for a tour group. How many ways are there to select a subset of 30 such tickets with the constraint that each of the 8 musicals appears on at least one of the tickets.

Fall 2008 Third Test for Tucker's AMS 301 course
1. Find the rook polynomial and give an expression for the number of matchings of 5 men (Rows) and 5 women (Columns) given the follow 8 conflicting pairs: (M1,W2),(M2,W1),(M2,W3),(M2,W5),(M3,W4),(M4,W2),(M4,W4),(M5,W5).
2. How many ways are there to form a committee of 12 mathematical scientists from a group of 15 mathematicians, 12 statisticians, & 10 operations researchers with at least one person of each different profession on the committee.
3. Find a rec. rel for an,k, the number of ways to select n hats from k diff. boxes of hats (all hats in a box are identical) with between 2 and 5 from each box.
4. a) Find a recurrence relation for an, the number of ways to give away n dollars by each successive day giving away $1 or $2 or $3.
b) Repeat part a) with the requirement that you cannot five away $3 one day, then $1 the next day followed by $2 the third day.
5. There are 10 different people. Each person orders three donuts, chosen from five types of donuts. How many wys are there to do this such that: (i) at least one person chooses all 3 donuts of the first type; (ii) at least one person chooses all 3 donuts of the second type, . . . , and (v) at least one person chooses all 3 donuts of the fifth type? NOTE: Two (or more) people may choose the same collection of three donuts.