After a recent reorganization, Mohammed is taking over responsibility for the manufacturing of odd numbers on the house sign number production line. As part of his scientific investigation into production levels, he wants to know how many three-digit numbers only contain odd digits. Calculate the answer for him.
In order to calculate how many three-digit numbers only contain odd digits, we could try listing them all out. This is called systematic listing. And it works nicely when there are just a very few number of options. In this case though, there’s likely to be quite a lot, so we’re going to recall a different method. This is called the counting principle or the product rule for counting. It says that to find the total number of outcomes for two or more events, we multiply the number of outcomes for each event together.
In our case, we have three events. The number of outcomes for event one will be the number of ways of choosing an odd number for our first digit. Event two is our second digit. So, the number of outcomes are the number of ways of choosing an odd number for our second digit. It follows, of course, that event three is the digit we get for our third digit. And so, the number of outcomes for event three is the number of ways of choosing an odd number for our third digit.
We should always ask ourselves as well whether the digit can be repeated. Well, yes, it makes complete sense that a house number would have repeated digits, such as 133 or 555. So, let’s look at the first event. We want to find the number of ways of choosing an odd digit for our first digit. Well, there are five odd digits we could choose. They are one, three, five, seven, and nine. So, there are five ways to choose our first digit. Since this number can be repeated, when we move on to the second digit, there are still five digits we can choose from.
Similarly, when we move on to the third digit, we can still choose from the same five odd numbers. The counting principle or the product rule for counting says that we can find the total number of three-digit numbers that contain odd digits by multiplying each of these together, or five cubed. That’s 125. And so, we see there are 125 three-digit numbers that we can find that only contain odd digits.