How many three-different-digit
numbers could be formed from the set of digits containing the elements one, two,
four, and nine?
Remember the fundamental
counting principle, sometimes called the product rule for counting, says that if
A is an event that has 𝑚 outcomes and B is an event that has 𝑛 outcomes, the
total number of outcomes of A and B together is the product of these. It’s 𝑚 times 𝑛. Now, in fact, we have three
possible events here. We have the event which is
choosing the first digit, the second event is choosing the second digit, and the
third event is choosing the third digit. But the counting principle
still holds, so we need to find out the number of outcomes we have for choosing
each digit and then multiply these together.
There are four possible digits
we can choose from. Those are one, two, four, and
nine. And so, it’s quite clear to us
that there are four different ways of choosing the first digit. Now, we’re told that they are
three-different-digit numbers. So, let’s think about how we
choose the second digit. Let’s say, for instance, the
first digit we chose was the number one. We can no longer use that. And so, there are three
different digits to choose from. The number of ways then of
choosing our second digit is three. Similarly, let’s move on to the
third digit. We’ve now already chosen two
numbers out of the set of four. And so, that means that there
can only be two numbers left to choose from.
And so, there are four ways of
choosing the first digit, three ways of choosing the second, and two ways of
choosing the third. The product rule for counting
or the counting principle tells us that we can find the total number of outcomes
by multiplying these together. That’s four times three times
two, which is equal to 24. There are 24
three-different-digit numbers that could be formed from the set of digits one,
two, four, and nine.