1. A restaurant chain advertised a special in which a
customer could choose one of the four appetizers,
one of 14 main dished, and one of five desserts. The
ad said that there were possible dinners. Was the ad
correct? Explain
2. In how many ways can we select a committee of four
women and three men from a group of six distinct
women and seven distinct men.?
3. How many strings can be formed by ordering the
letters ”ALGEBRAICALLY”.
4. In how many ways can 12 distinct books be divided
among three students if the first person students gets
five books, the second three books, and the third two
books?
5. Find an explicit solution to each recurrence relation:
(a) an = 3an−2 − 2an−4,
(b) 3an = 48an−4
6. Solve the recurrence relation an =
s
an−2
an−1
with ini-
tial conditions a0 = 8, a1 = 1
2√2 by taking natural
logarithm of both sides and making the substitution
bn = ln an
rence relations;
A(m , 0) = A(m – 1, 1)
and
A(m, n) = A(m – 1, A(m, n – 1))
,
m, n = 1, 2, · · ·(1)
and the initial conditions
A(0, n) = n + 1 , n = 0, 1, · · ·,(2)
Determine the value of
(a) A(1, 1)
(b) A(2, 2)
(c) A(3, 2)
8. Refer to the Catalan numbers defined by the recur-
rence relation (n + 2)Cn+1 = (4n + 2)Cn, for n ≥ 0
and the initial condition C0 = 1. Find
(a) C1
(b) C2
(c) C7
2
represented by each adjacency matrix with vertices:
a, b, c, d
A =
0 1 1 1
1 0 1 0
1 1 0 1
1 0 1 0
10. Draw an undirected graph and a directed graphs
represented by each adjacency matrix with vertices:
a, b, c, d
B =
0 1 1 0
0 0 1 0
0 0 0 1
1 0 0 0
11. Draw an undirected graph represented by the inci-
dence matrix with vertices:
v1, v2, v3, v4 and edges: e1, e2, e3, e4, e5
C =
1 0 0 1 1
1 1 0 0 0
0 1 1 0 1
0 0 1 1 0
12. Draw a directed graphs represented by the incidence
matrix with vertices:
v1, v2, v3, v4 and edges: e1, e2, e3, e4, e5
D =
1 0 0 −1 1
−1 1 0 0 0
0 −1 1 0 −1
0 0 −1 1 0
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defined by R1 = {(1, x), (1, y), (2, x), (3, x)}; and let
R2 be the relation from Y to Z = {a, b, c}
defined by R2 = {(x, b), (y, b), (y, a), (y, c)}; find
(a) The matrix A1 of the relation R1 (relative to the
given orderings).
(b) The matrix A2 of the relation R2 (relative to the
given orderings).
(c) The matrix A1A2.
(d) Use the result of part (c) to find the relation
R2 ◦ R1.
14. Let R1 = {(x, y) : x|y} be a relation from X to Y ;
R2 = {(y, z) : z < y} be the relation from Y to
Z; ordering of X and Y : 2, 3, 4, 5 and ordering of
z : 1, 2, 3, 4
(a) The matrix A1 of the relation R1 (relative to the
given orderings).
(b) The matrix A2 of the relation R2 (relative to the
given orderings).
(c) The matrix A1A2.
(d) Use the result of part (c) to find the relation
R2 ◦ R1.
4
15. How many different car licensed plates can be
constructed if the licenses contain three letters fol-
lowed by two digits if
(a) repetitions are allowed?
(b) repetitions are not allowed?
16. In how many different ways can 14 horses finish in
the order Win, Place, Show?
17. Assume that
(1−x)Σ∞
n=2n(n−1)anxn−2 + xΣ∞
n=1nanxn−1 = Σ∞
n=0anxn
for all x.
(a) Show that the coefficients an is given by the
recurrence relation an+2 = (n + 1)nan+1 − (n − 1)an
(n + 1)(n + 2) ,
for n ≥ 1.
(b) Let a0 = 2, find a2, a3, a4, a5 and a6.
18. Assume that
Σ∞
n=2n(n − 1)anxn−2 = 2xΣ∞
n=1nanxn−1 – λΣ∞
n=0anxn
for all x.
(a) Show that the coefficients an is given by the
recurrence relation an+2 = (2n − λ)an
(n + 2)(n + 1),
for n ≥ 0.
(b) If λ = 2 and a0 = 1, find a2, a3, a4, a5 and a6.
19. Defined each of the mathematical system:
Axiom, definition, undefined term, theorem,
proof, lemma, corollary, and direct proof
