Math 267 Prof. Brick
section 1 Spring 02
Discrete Math Exam 2
Do the problems in order in your bluebook. Show your work.
1. How many ternary strings of length 120 are there ?
2. A jar contains marbles colored blue, green, red, and yellow. Assuming you are blind-
folded, how many marbles must you take to be sure to get 7 of the same color ? Explain
it using the pigeonhole principle.
3. Give a combinatorial proof that C(n, k) = C(n, n −k).
4. An exam consists of 40 true-false questions. Ten of the questions have “true” as the
answer. How many answer keys are possible.
5. You flip a coin 4 times. Show that the events “the first three flips are heads” and “the
4th flip is heads” are independent.
6. Find the probability of getting a full house in 5-card poker.
7. Find the recurrence relation for the number of ways of climbing nstairs, assuming you
can take them one or two or three stairs at a time.
8. Consider the set of subsets of the set {1,2}and the partial ordering given by inclusion.
Draw the Hasse diagram.
9. Let f:X→Ybe a function. Define a relation Ron Xby x1Rx2iff f(x1) = f(x2).
Prove that Ris an equivalence relation.