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Dr. A. Betten Fall 2009 MATH 501 Introduction to Combinatorial Theory Assignment # 4 Problem # 15 Find the number of integers between 1 and 1000, inclusive, that are not divisible by 5, 6 and 8. Problem # 16 A box contains 7 balls: blue one, blue two, blue three, red one, red two, green one and green two. Three of the balls are to be selected at random. How many such subsets contain at least two blue balls? Problem # 17 For each of the following expressions, list the set of all formal products in which the exponents sum to 4. (a) (1 + x + x2 )2 (1 + x)2 (b) (1 + x + x2 + x3 + x4 )3 (c) (1 + x2 + x4 )2 (1 + x + x2 )2 (d) (1 + x + x2 + x3 + · · · )3 Problem # 18 How many ways are there to give change for $1 using pennies, nickels, dimes and quarters? Use a generating function approach. Problem # 19 A tree is a connected graph without cycles. Show all trees on 4, 5, 6 vertices (up to isomorphism). Problem # 20 A quadrangle in a projective plane is a set of 4 points, no three on a line. Count the number of quadrangles in a projective plane of order q. Problem # 21 A riffle shuffle of a deck of 2n cards is a permutation of the form ! " 1 2 3 4 5 · · · 2n − 1 2n 1 n + 1 2 n + 2 3 ··· n 2n The order of a permutation π is the least positive integer a such that π a = id (π a is π repeated a times). Find the order of the riffle shuffle for 2n = 10, 12, 14, 16, 24. Bonus Problem Problem # 22 Find a graph whose automorphism group is the cyclic group of order 3 (exactly, not as a subgroup).