Discrete Mathematics: Revision Questions
Q1. Find the number of different arrangements of the letters in the word: MISAPPREHENSION.
Of these arrangements,
(a)
(b)
(c)
(d)
how
how
how
how
many
many
many
many
have all the vowels together?
start and end with N?
start or end with N?
have all the letters in alphebetical order?
Q2. A committee of 4 is to be chosen from 6 married opposite-sex couples. In how many ways can this be done?
Of the possible committees how many contain exactly 2 women?
Of the possible committees how many contain at least 2 women?
Of the possible committees how many contain no married couple?
n−1
Q3. (a) Prove that k nk = n n−1
(b) Prove that nk = n−1
k−1 .
k−1 +
k .
Q4. (a) How many (binary) bit strings are there of length 8?
How many of these have weight 3?
(b) A ternary string is a sequence of 0s, 1s, and 2s. How many ternary strings of length 17 are there?
How many of those strings contain exactly eight 0s, four 1s, and five 2s?
How many ternary strings of length 17 contain an odd number of 1s?
Q5. How many non-negative integer solutions are there to the equation x1 + x2 + x3 + x4 + x5 < 11, if there are
no restrictions?
How many solutions are there if x1 > 3?
How many solutions are there if each xi < 3?
Q6. Students in an Indiscreet Mathematics class work together in groups of 5 for an assignment. The group is
given a score, which they divide up, according to the amount of work each did, to get their individual scores.
Aoife, Brian, Conor, Declan, and Eimer worked together, and got a score of 20.
Q7.
(a) How many ways can their scores be assigned?
(b) The were given scores (respectively) of 2, 4, 6, 8 and 0. The lecturer entered these scores, but assigned
them all to the wrong people. How many ways can this happen?
X
(a) Prove that
deg(v) = 2|E| for any graph G = (V, E). Deduce that the number of edges in the complete
v∈V
n
graph on n vertices is equal to
.
2
(b) Prove that if a connected planar graph has v vertices, e edges, and f faces, then v − e + f = 2. Use this
to show that K3,3 is not planar.
Q8. Determine the chromatic number of each of the following graphs, and give a corresponding colouring.
(a) Ga =
(b) Gb =
(c) Gc =
Q9. Explain the terms Eulerian path and Eulerian circuit.
For each of the following graphs, determine if it has an Eulerian path and/or Eulerian circuit. If so, give an
example; if not, explain why.
(a) G = (V, E) with V = {a, b, c, d, e, f } and E = {a, b}, {a, c}, {a, d}, {a, f }, {b, c}, {b, d}, {b, e}, {c, e}, {c, f }, {d, e}, {d, f
(b)
g
c
i
j
f
a
e
d
b
Q10.
h
(a) Show that if T is a tree with e edges, then it has e + 1 vertices.
(b) Show that if T is an acyclic graph with v vertices, and e = v − 1 edges, then it is a tree.