How many three-digit numbers are there ____.

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Detailed Solution

Here, we must state how many three-digit numbers there are. We need three places—ones, tens, and hundreds—to create a three-digit number. The number system that we use has a total of

10

digits, i.e

. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

. These digits can be added to our three places to produce the three-digit number. Any of the ten’s digits may be placed at one's location. Therefore, there are a total of 10 possible ways to choose digits for a location.
Any tens digit may be placed in the tens place. The total number of ways to choose digits for the tens place will therefore be

9 .

Let's now examine hundreds of locations.
As we can see, we are only able to place

9

digits instead of all

10

because

0

cannot exist in hundreds of locations. This causes our number to become a two-digit number, which is not needed. The total number of digits that can be used in the hundreds place is therefore

9

\frac{9 ways}{Hundreds} \frac{10 ways}{Tens} \frac{10 ways}{Ones}

.
We simply need to multiply all the ways to make ones, tens, and hundreds in order to find the total ways to make numbers. As a result, the total becomes