DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1992, 2008.
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Chapter 15
NUMBER OF SOLUTIONS
OF A LINEAR EQUATION
15.1. Introduction
Example 15.1.1.
Suppose that 5 new academic positions are to be awarded to 4 departments in the
university, with the restriction that no department is to be awarded more than 3 such positions, and
that the Mathematics Department is to be awarded at least 1. We would like to ﬁnd out in how
many ways this could be achieved. If we denote the departments by
M,P,C,E
, where
M
denotes
the Mathematics Department, and denote by
u
M
,u
P
,u
C
,u
E
the number of positions awarded to these
departments respectively. Then clearly we must have
u
M
+
u
P
+
u
C
+
u
E
= 5
.
(1)
Furthermore,
u
M
∈ {
1
,
2
,
3
}
and
u
P
,u
C
,u
E
∈ {
0
,
1
,
2
,
3
}
.
(2)
We therefore need to ﬁnd the number of solutions of the equation (1), subject to the restriction (2).
In general, we would like to ﬁnd the number of solutions of an equation of the type
u
1
+
...
+
u
k
=
n,
where
n,k
∈
N
are given, and where the variables
u
1
,...,u
k
are to assume integer values, subject to
certain given restrictions.
Chapter 15 : Number of Solutions of a Linear Equation
page 1 of 13