In this video, we will look at three examples
of the fundamental counting principle,
which states if there are m ways to do one thing
and n ways to do another,
then there are m times n ways of doing both.
For the first example, standard automobile license plates
in a country display two numbers,
followed by two letters, followed by three numbers.
How many different standard plates
are possible in this system?
Assume repetition of letters and numbers are allowed.
So the license plate has seven digits
where the first two digits are numbers
which we'll label as N,
followed by two letters, we'll label these L,
followed by three numbers, we'll label these N.
The numbers can be any digit from zero to nine.
So there are 10 choices for the numbers.
The letters can be any letter in the alphabet A through Z,
and therefore, there are 26 choices for the letters.
The counting principle states
if we multiply the number of ways
we can select the numbers and letters,
we can determine the number of different license plates.
So there are 10 choices for the first number,
10 choices for the second number,
the letters have 26 choices.
And again, the numbers have 10 choices.
Notice how we have five factors of 10 and two factors of 26.
The product can be written as 10 to the fifth
times 26 squared.
And now let's go to the calculator.
We have 10 raised to the fifth, right arrow,
times 26 squared, enter,
which gives us 67,600,000 license plates.
Next, you are taking a quiz
that has eight multiple choice questions.
If each question has five possible answers,
how many different ways are there to answer the test?
So we'll call this question one,
question two through question eight.
The counting principle states if we multiply the number
of choices for each of these questions, we can determine
the number of different ways to answer the test.
And because each question has five possible answers,
we have five times five times five times five
times five times five times five times five.
We have eight factors of five,
which is equal to five raised to the eighth power.
So going to the calculator.
Five raised to the power of eight is equal to 390,625.
There are 390,625 different ways
to answer the test.
A gate has a standard keypad
with the digits zero through nine.
How many possible code combinations are there
if the code is six digits long?
And we assume no restriction on repetition.
So, because there are six digits,
we'll call this digit one, digit two, and so on.
The counting principle states
we can determine the number of six-digit codes
by determining the product of the number of ways
we can select each digit.
Because the digits are zero through nine,
there are 10 choices for each digit,
which gives us 10 times 10 times 10
times 10 times 10 times 10,
which is equal to 10 to the sixth,
which is going to be one followed by six zeroes,
which is 1,000,000.
There are 1,000,000 possible codes.
I hope you found this helpful.