Test Grade Guidelines

These are approximate grade middles. The letter grades are only to give you an idea of how well you're doing. I don't add letter grades. I add up your scores to get the total of test points.

Test 1 (10/18)

Test 2

Final Exam

Quizzes and Solutions

• Quiz 1 (8/30): Show that an m × n chessboard has a perfect cover by dominoes if and only if m or n is even. [This is Exercise 1.1.]
• Quiz 2: (Not graded.)
• Quiz 3 (F 10/14): In the partially ordered set P(X), where X is a finite set, describe the cover relation. I.e., describe when set A ∈ P(X) covers set B ∈ P(X). [This is Exercise 4.42.]
  Solution: A covers B when B = A − {x} for some element x ∈ A.
• Quiz 4 (Tu 11/1):
  1. What is the sequence whose generating function is (*) x + 3x² + 5x³ + 7x⁴ + ···?
  2. What is the sequence whose exponential g.f. is (*)?
  Solutions:
  1. (0, 1, 3, 5, 7, ...). (Congratulations if you wrote a_n = 0 if n = 0 and 2n+1 if n > 0, but it was not required.)
  2. (0, 1, 3·2!, 5·3!, 7·4!, ...) = (0, 1, 6, 30, 168, ...). (Congratulations if you wrote a_n = 0 if n = 0 and (2n+1)n! if n > 0, but it was not required.)
• Quiz 5 (W 11/9): "Simplify" (x+1)/(x³−3x³−x+3) using partial fractions.
  Solution: The first step is to factor the denominator. Notice the repeated coefficients 1, 3, 1, 3? That's a strong hint. Result:
              x³−3x³−x+3 = (x−3)(x²−1), and take it from there.
• Quiz 6 (F 11/11):
  1. Write a precise definition of (x)_n for n ∈ Z_≥0.
  2. Simplify (1/2)_n, for n ∈ Z_≥0, as far as possible.
  Solutions:
  1. (x)_n = x(x−1)(x−2)···(x−n+1), and it's good to mention the special case (x)₀ = 1.
  2. (1/2)₀ = 1;
     (1/2)₁ = 1/2;
     (1/2)_n = (−1)ⁿ⁻¹ (1)(3)···(2n−3)/2ⁿ, for n ≥ 2 (and n = 1 if you like empty products).
     Further "simplification" (dubious word) is possible but this is quite satisfactory.
• Quiz 7 (W 11/30):
  1. What is the number of lattice paths from (0,0) to (p,q) using steps Right and Up? (Assume p, q ≥ 0.)
  2. What is the sequence h_n whose generating function is 1/(1+3x)⁴?
  Solutions:
  1. binom(p+q, p), or binom(p+q, q) (they're equal).
  2. h_n = binom(n+3, 3)·(−3)ⁿ. Read this off from the general formula (5.22) at the bottom of page 147.
• Quiz 8 (F 12/2): Let h_n = binom(n+6, 6)·(−3)ⁿ. Simplify the generating function of h_n as far as possible.
  Solution: The 6 gives it away: 1/(1+3x)⁷.
• Quiz 9 (Tu 12/6): In F₂₇, with polynomial x³+2x+1=0, evaluate x⁵.
  Solution: 27=3³ so the polynomial has coefficients in Z₃ and we need a polynomial of degree 3 (the exponent of 3³), which is what we have. First, solve for x³=−(2x+1)=x+2. Next, simplify:
        x⁵ = x²(x³) = x²(x+2) = x³+2x² = (x+2)+2x² = 2x²+x+2,
  which has degree < 3. Solved!