Honors Introduction to Discrete Structures
COT 3100H Exam #2: Counting, Probability, Induction, Number Theory
Date: 3/20/08
Name: _______________________
Directions: Please leave the answers to each question in terms of powers, factorials
and combinations.
1) (15 pts) Each of the following questions will deal with permutations of the letters in
"HILLARYCLINTON"
a) How many permutations are there utilizing all of the letters in HILLARYCLINTON?
b) How many combinations of distinct consonants are there from the letters in
HILLARYCLINTON? (Y is a consonant.)
c) How many permutations of the letters in HILLARYCLINTON have two vowels
adjacent to one another? (A, E, I, O, and U are vowels.)
2) (5 pts) Given n sets of parentheses (n open parentheses and n close parentheses), the
probability that a random permutation of them forms a valid arrangement of parentheses
is 0.01. What is the value of n?
3) (15 pts) A store has 20 parking spaces in a row. After 15 people have parked
randomly, what is the probability that two (or more) of the five open spaces are adjacent?
4) (10 pts) The probability that Dwight Howard posts a double-double in a single game is
90%. If he does post a double-double, the Magic win 75% of the time. If he does NOT
post a double-double, the Magic win 50% of the time. Given that the Magic lost a game,
what is the probability that Dwight Howard posted a double-double?
5) (10 pts) Let S be the set of all of the divisors of 64. Let T contain the reciprocals of all
(2 5  1)(35  1)
the values in S. Explain why the sum of the values in T is
.
2 534
6) (15 pts) Find all integer solutions to the equation 602x + 217y = 7.
n
0
 1 0
 1
   n
 for all positive
7) (10 pts) Use induction to show that 
n

1
2

2

1
2




integers n.
8) (15 pts) Prove that 49 | (23n – 7n – 1) for all non-negative integers n.
9) (5 pts) On what day of the week did Palm Sunday occur? ______________________
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