CMSC 203 - Discrete Structures - Spring 1999 - Exam 3
1. Suppose I have a collection of 30 Math books, 20 Chemistry books, 10 Biology books,
and 40 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 adjacent?
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?
e. How many ways can I select 12 books if all the books are distinct and I want 3 of
each type?
2. Suppose 10 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?
3. Verify:
4. Fill in the blanks for the first 10 rows of Pascalโs Triangle:
Row 0: 1
Row 1: 1 1
Row 2: 1 _____ 1
Row 3: 1 _____ _____ 1
Row 4: 1 _____ _____ _____ 1
Row 5: 1 _____ _____ _____ _____ 1
Row 6: 1 _____ _____ _____ _____ _____ 1
Row 7: 1 _____ _____ _____ _____ _____ _____ 1
Row 8: 1 _____ _____ _____ _____ _____ _____ _____ 1
Row 9: 1 _____ _____ _____ _____ _____ _____ _____ _____ 1
5. Find the next 5 terms in the recurrence relation:
sn = snโ1snโ3 โ snโ2 when s0 = 1, s1 = 0 and s2 = (โ1)
6. Use the Method of Iteration to find a general solution to the recurrence relation:
sn = 3snโ1 + 2, when s0=1.
7. Find the general solution to the recurrence relation whose characteristic polynomial
has roots 3,3,3,3,3,(โ2),(โ2),(โ2).
8. Find the general solution to the 2nd order, linear, homogeneous recurrence relation
with constant coefficients: sn = 2snโ1 + 15snโ2.
9. Find the particular solution to the recurrence relation whose general solution is
sn = A4n + B(โ3)n, when s0=7 and s1=14.
n
k
โโ
โโ n1โ
k1โ
โโ
โโ
n1โ
k
โโ
โโ
+=
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