Archived Notes on Grading: Prelim II

Please refer to our guidelines for induction proofs.

In general, we are very picky when we grade induction proofs.

Our approach to this question exploits the fact that the number of solutions to (1) under the constraint D₁ is simply four times the number of solutions to (1) under the constraint D₂. Please note that this shortcut is valid only because D₁'s four disjuncts are pairwise disjoint (i.e., only because you can't possibly satisfy (1) when x_i >= 10 and x_j >= 10 and i != j). [If the disjuncts hadn't been mutually exclusive, we would have had to deal with the 27 intersection terms in the full inclusion-exclusion formula (ouch!).]

If you see "disjointness?" on your paper, then you neglected to address / mention the disjointness issue described above.

If you see "<= constraints?" on your paper, then you computed the number of solutions to (1) but did not enforce the x_i <= 9 constraints. [Without the <= 9 constraints, you're counting solutions like "0 0 0 12" (i.e., x₁ = 0, x₂ = 0, x₃ = 0, and x₄ = 12) that do not correspond to a valid, 4-digit decimal number.]

[If you're puzzled by the notation used above--(1), D₁, D₂, etc.--please see our solution to this question.]

Especially in probability and counting exercises, it is very easy to get things wrong; you should always explain what you do. A bunch of numbers doesn't prove anything if nobody can tell what you are doing. We gave more credits to wrong answers with explanations than to the (presumably) same wrong answers without explanations.