Math 150 Final Exam December 17, 2009

Choose 6 from the following 9 problems. Circle your choices: 1 2 3 4 5 6 7 8 9 You may do more than 6 problems in which case your unchosen problems can replace your lowest one or two problems at 4/5 the value as discussed in class.

1.) Use a generating function to find the number of combinations of {3·a, 1·b, ∞·c} which have exactly 10 elements.

2.) Use inclusion-exclusion to to find the number of combinations of {3·a, 1·b, ∞·c} which have exactly 10 elements.

3.) Use Theorem 14.2.3 to determine the number of nonequivalent colorings of the corners of a rectangle that is not a square with the colors red and blue. Do the same with p colors.

4.) Solve the following homogeneous recurrence relation: h_{n} = 4h_{n−1} + 5h_{n−2}.

5a.) Use the binomial theorem to prove that 2^{n} = Σ_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ).

5b.) Use a combinatorial argument to prove that 2^{n} = Σ_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ).

6a.) Suppose S is a set of n integers. Show that given any d < n, there exists x, y ∈ S such that x ≠ y and d divides x − y.

6b.) Show that the Ramsey number r(3, 3) > 5.

7a.) ( \begin{matrix}1.6 \\ 3 \end{matrix} ) =

7b.) Let f = ( \begin{matrix}1234 \\ 1342 \end{matrix} ).

Let c : \{1, 2, 3, 4\} → \{red, blue\}, c(i) = \{ \begin{matrix}red & i = 0 mod 2 \\ blue & otherwise \end{matrix}

f^{−1} = f ∗ c = 8a.) The inversion sequence for the permutation 1324 is 8b.) Define the relation ≤ on R × R by (a, b) ≤ (c, d) if a < c or if a = c, b ≤ d. Show the relation ≤ on R × R is anti-symmetric. 9.) Let A_{1} = {1, 2}, A_{2} = {3, 5, 6} , A_{2} = {4} and let X = {1, 2, 3, 4, 5, 6} . Define a relation R on X by xRy if there exists A_{i} such that x, y ∈ A_{i}. Draw R as a subset of X × X. Determine which of the following properties hold for R (Prove it).

Is R reflexive? Is R irreflexive? Is R symmetric? Is R antisymmetric? Is R transitive? Is R an equivalence relation? If so, use R to partition X into its equivalence classes. Is R a partial order?