Third Test of Tucker's AMS 301 course
Fall 2008 Third Test for Tucker's AMS 301 course
TEST A
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 peopple. 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.


Fall 2007 Third Test of 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 following 9 conflicting pairs: (M1,W1), (M2,W4), (M3,W2), (M3,W5), (M4,W1), (M4,W4), (M5,W2), (M5,W3), (M5,W5).
2. (10 pt) How many ways are there to arrange the 26 letters of the alphabet in a row such that none of the following words are formed by consecutive letters in the arrangement: INCH, LOST, or THIN?
3. (4 pt) Find a rec. relation for an,m, the number of ways to select n identical balls from m different boxes with between 2 and 5 balls (2 or 3 or 4 or 5) from each box.
4a) (5 pt) Find a recurrence relation for an, the number of sequences of red flags, white flags and yellow flags along an n-foot high flagpole if red and white flags are 1 foot tall and yellow flags are 2 feet tall.
b) ( 6 pt) Repeat part a) with the constraint that no red flag can be followed by a yellow flag.
5. (15 pt) There are balloons of 7 different colors. Six (distinct) people each select 2 balloons (possibly of the same color)- each person's selection is different. How many outcomes are there for the 6 people's selections if each color of balloon is used at least once by one (or more) people?
5' (alternative for 10 pt) There are 18 students, three (distinct) students each from 6 different high schools. There are 6 admissions officers, one from each of 6 colleges. Each of the officers successively picks a subset of three of the 18 students to go to their college (once a student is chosen, another college cannot chose him later). How many ways are there to do this so that no officer picks 3 students from the same high school?

Fall 2006 Third Test of 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 following 8 conflicting pairs: (M1,W1), (M1,W4), (M2,W3), (M2,W5), (M3,W5), (M4,W2), (M4,W4), (M5,W5).
2. How many ways are there to arrange the letters in the word MISSISSIPPI so that the 4Is are not consecutive, the 4 Ss are not consecutive, and the 2 Ps are not consecutive?
3. For a recurrence 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. a) Find a recurrence relation for An, the number of sequences forms by a's, b's, and c's with the subsequence aa not allowed (no consecutive a's).
b) Repeat part a) but change the forbidden subsequence to abc.
5. A swim suit designer produces a collection of 15 versions for a certain type of bikini (the bikinis differ only in their colors). There are 6 colors to choose from for the top part and the bottem part of the bikini, and the top and bottom parts must be DIFFERENT colors. How many collections (subsets) of 15 versions are possible if each color appears in at least one of the versions?