CMSC 203 - Discrete Structures - Fall 2000 - Exam 3
1. Suppose I have a collection of 27 Math books, 18 Chemistry books, 22 Biology books,
and 25 Geology books.
a. How many ways can I arrange all these books on a shelf if all the books are distinct?
b. How many ways can I arrange all these books on a shelf if all the books are distinct
and I want the books of each category to be grouped together?
c. How many ways can I select 10 books if all the books of each type are the same?
d. How many ways can I select 10 books if all the books of each type are the same and
I want at least 2 of each type?
2. How many orderings are there of the letters of the words:
a. CHEMISTRY b. MATHEMATICS
3. Suppose 15 people go to eat at a restaurant.
a. How many ways can they arrange themselves around a round table?
b. How many ways can they arrange themselves around a round table if a certain pair of
people cannot sit adjacent to one another?
4. Verify:
5. If the a row of Pascalโs Triangle is โ1, 5, 10, 10, 5, 1โ what is the next row?
6. Find s7 in the recurrence relation:
sn = (snโ1)(snโ3) โ (snโ2) when s0 = 0, s1 = 1 and s2 = (โ1)
7. Use the Method of Iteration to find a general solution to the recurrence relation:
sn = 3snโ1 + 2, when s0 = 5.
8. Find the general solution to the recurrence relation whose characteristic polynomial
has roots 3,3,3,(โ4),(โ4),(โ4),5,5,5,(โ6),(โ6),(โ6),7,7,7 .
9. Find the general solution to the 2nd order, linear, homogeneous recurrence relation
with constant coefficients: sn = 2snโ1 + 63snโ2.
10. Find the particular solution to the recurrence relation whose general solution is
sn = A4n + B(โ3)n, when s0 = 5 and s1 = 34.
n
k
โโ
โโ n1โ
k1โ
โโ
โโ
n1โ
k
โโ
โโ
+=
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