Counting Problems with Solutions

This page presents the counting principle through clear explanations and fully worked counting problems.

The Counting Principle

A student must choose one physics course, one science course, and one mathematics course.

The total number of possible course selections is:

\[ N = 3 \times 2 \times 2 = 12 \] Tree diagram showing all possible course selections

In general, if events \(E_1, E_2, E_3, \ldots\) can occur in \(n_1, n_2, n_3, \ldots\) ways respectively, then the total number of outcomes is:

\[ N = n_1 \times n_2 \times n_3 \times \cdots \]

Problem 1

A customer chooses:

How many different computer systems are possible?

Solution

\[ N = 4 \times 2 \times 4 \times 3 = 96 \] Possible choices for computer components

Problem 2

Telephone numbers have 9 digits. The first two digits are fixed (03). The remaining 7 digits cannot begin with 0.

Solution

\[ N = 1 \times 1 \times 9 \times 10^6 = 9{,}000{,}000 \] Choices for telephone number digits

Problem 3

A student chooses one mathematics book from 6 options, one chemistry book from 3 options, and one science book from 4 options. In how many different ways can the student select one book from each subject?

Solution

\[ N = 6 \times 3 \times 4 = 72 \]

Problem 4

There are 3 roads from city A to B and 2 roads from B to C. How many possible roads are there from city A to city C?

Solution

\[ N = 3 \times 2 = 6 \]

Problem 5

A man has a wardrobe consisting of 3 suits, 4 shirts, and 5 pairs of shoes. In how many different ways can he choose an outfit?”

Solution

\[ N = 3 \times 4 \times 5 = 60 \]

Problem 6

An ID card has 5 digits. How many different IDs can be made

  1. when digits may repeat, and
  2. when digits may not repeat?

Solution

a) Repetition allowed

\[ N = 10^5 = 100{,}000 \] Digit choices with repetition

b) Repetition not allowed

\[ N = 10 \times 9 \times 8 \times 7 \times 6 = 30{,}240 \] Digit choices without repetition

Problem 7

A licence plate has 3 letters followed by 4 digits. How many different licence plates are possible?

Solution

\[ N = 26^3 \times 10^4 = 175{,}760{,}000 \]

Problem 9

A coin is tossed three consecutive times. Determine the total number of possible outcomes. \[ N = 2^3 = 8 \]

Problem 10

When two dice are rolled, how many possible outcomes are there? \[ N = 6 \times 6 = 36 \]

Problem 11

A coin is tossed and a six-sided die is rolled. How many possible outcomes are there? \[ N = 2 \times 6 = 12 \]

Further Reading

Elementary Statistics and Probability