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: (mi, w1), (m2,w4), 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 nch, 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?