THE UNIVERSITY OF MANITOBA

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1 THE UNIVERSITY OF MANITOBA MATHEMATICS 3400 (Combinatorics I) March 4, 2011 MIDTERM EAM 2 hours (6-8 PM) EAMINER: R. Craigen Total Marks: 100 NAME: STUDENT NUMBER: INSTRUCTIONS Write your name and student number above Read all questions carefully before answering them. Show all necessary work. How well you follow instructions, neatness, and presentation may all affect your grade. If you continue your work outside the space alloted, clearly indicate where to find an answer, or portion thereof, if it is out of order. Perform any easy simplification of your answers. Large or complicated numerical answers, such as 100! may be left as the most simple formula that is easy to reach by hand. Scientific or graphing calculators and a one-sided crib sheet are permitted, but unnecessary, in this exam. To show that you read these instructions, and for two bonus marks, circle your student number above. Marks for each question are indicated in the column on the left. [20] 1. True or False? Answer (T or F) in left margin. Statements that are meaningless (for us) or nonsense are considered false. (a) (1 + 2) 1 2 = ( 1 ) 2 2 k. k k 0 (b) The number of ways k identical pigeons can roost in n distinguishable holes is ( ) n+k 1 k. (c) A generating function must have radius of convergence greater than 0. (d) At christmas every student in a class of 100 students puts their name into a jar. It is shaken to randomize it, and each student draws out one of the names for the purpose of gift giving. It is more likely than not that some student draws their own name. (e) The rook polynomial for a board consisting of two disjoint subboards is the sum of the rook polynomials for the two subboards. (f) Rook polynomials are not generating functions. (g) ( ) ( 7 12 = 18 12). (h) If A is a set of a elements and B is a set of b elements, then the number of functions from A B to A is (ab) a. (i) There are more onto functions from a 7-set to a 5-set than one-to-one functions from a 5-set to a 7-set. (j) The number of paths from (0, 0, 0) to (a, b, c) in the integer lattice, composed only of individual steps taking a vertex (x, y, z) to (x + 1, y, z), (x, y + 1, z) or (x, y, z + 1), is (a+b+c)! a!b!c!.

2 [30] 2. Compute, and simplify (as far as reasonable), the following numbers (3 marks each): (a) P (10, 3) (b) The number of 5-card hands in which there are more red cards than black, and which include a 3-of-a-kind (i.e., not part of a 4-of-a=kind). (c) The number of derangements of 7 objects. (d) The number of paths through the integer lattice from A(12, 3) to B(4, 13), using only moves taking a point (x, y) to (x, y + 1) or (x 1, y). (e) [x 10 ]( 1 1 x 1 1 2x ) 2

3 (f) The number of 13-letter words that can be formed with the multiset of letters from the word COMBINATORICS. (g) The coefficient of x 7 y 2 in (3 + 2x 5y) 11 ; (h) The number of distinct divisors of 10!. (i) The number of ways of giving 12 red and 7 green jellybeans to 3 children such that the youngest receives at least two of each colour. (j) The number of seating arrangements for five couples around a circular table such that everyone sits beside their spouse. 3

4 [15] 3. Answer each briefly. (a) How many positive integers are less than 1000 and divisible by none of 4, 9 or 6? (b) Give, in closed form, a generating function for which the coefficient for x n is the number of solutions to the equation w + 2x + 3y n = 0, assuming that each of w, x, y, z is a nonnegative integer, w, x 3, and y 5. (c) Find the coefficient of x 3 in the generalized binomial expansion of x. 4

5 [6] 4. Let s n = 1 1! + 2 2! + 3 3! + + n n!. Use mathematical induction to prove that s n = (n + 1)! 1, for all n 1. [8] 5. Prove that, among any set of five integer lattice points in the plane, there are two such that the midpoint of the segment having those two as endpoints is also a lattice point. (HINT: Pigeon hole principle.) 5

6 [3] [2] [4] 6. (a) Find a generating function for n 3. (b) Find a generating function for the sum of the first n cubes. (c) Derive the (known) formula for the sum of the first n cubes from this generating function. 6

7 [12] 7. In each of the following chessboards, an marks each forbidden square: A = B = (a) Find the rook polynomial of board A. (b) In how many ways can 5 mutually nonattacking rooks be placed on board B? 7

8 BONUS: DO ONLY ONE of the following questions, for extra credit ( 10%). If you attempt more than one, indicate clearly which you wish marked. There may be more work than you can complete, but any significant progress on a bonus question will receive some credit; marks will be assigned according to the quality and completeness of your response, taking into account the difficulty of the question (more challenging questions worth more credit!). A: If f(x) = k 0 a k x k is a formal power series and a 0 0, prove that 1 f(x) is also expressible, uniquely, as a formal power series (where the reciprocal is taken to have the only possible reasonable meaning). How does the reciprocal of f(x) = 1 + 2x + 3x 2 + begin? B: In how many ways can 8 identical bishops be placed onto a 5 5 chessboard so that no two of them are on a common diagonal line? C: How many paths (with the usual description) are there in the integer grid, from (0, 0) to (10, 15), that do not pass through any of the points (2, 2), (3, 5) or (7, 8)? TOTAL MARKS: 100 8