Counting Principle

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Counting Principle
Let us start by introducing the counting principle using an example. A
student has to take one course of physics, one of science and one of
mathematics. He may choose one of 3 physics courses (P1, P2, P3), one
of 2 science courses (S1, S2) and one of 2 mathematics courses (M1, M2).
In how many ways can this student select the 3 courses he has to take?
Let us use a tree diagram that shows all possible choices. The first column
on the left shows the 3 possible choices of the physics course: P1, P2 or
P3. Then the second column shows the 2 possible choices of the science
course and the last column shows the 2 possible choices for the
mathematics course. The different ways in which the 3 courses may be
selected are:
(P1 S1 M1), (P1 S1 M2), (P1 S2 M1), (P1 S2 M2)
(P2 S1 M1), (P2 S1 M2), (P2 S2 M1), (P2 S2 M2)
(P3 S1 M1), (P3 S1 M2), (P3 S2 M1), (P3 S2 M2)
The total number of choices may be calculated as follows:
Let n1 be the number of choices of the physics course, here n1 = 3. Let n2
be the number of choices of the science course, here n2 = 2. Let n3 be the
number of choices of the mathematics course, here n3 = 2. It is clear from
the tree diagram above that the total number N of choices may be
calculated as follows:
N = n1 × n2 × n3
= 3 × 2 × 2 = 12
Using the above problem, we can generalize and write a formula related to
counting as follows:
"If events E1, E2, E3 ... can occur in n1, n2, n3 ... different ways
respectively, the number of ways that all events can occur is equal to
n1 × n2 × n3 ..."
Problem 1
To buy a computer system, a customer can choose one of 4 monitors, one
of 2 keyboards, one of 4 computers and one of 3 printers. Determine the
number of possible systems that a customer can choose from.
Solution to Problem 1

A customer can choose one monitor, one keyboard, one computer
and one printer. The diagram below shows each item with the
number of choices the customer has.

Using the counting principle used in the introduction above, the
number of all possible computer systems that can be bought is given
by
N = 4 × 2 × 4 × 3 = 96
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