Answer:
48
Step-by-step explanation:
Solution 1:
There are 5 even digits: 0, 2, 4, 6, and 8, and ways to arrange any three of them in order: 5 ways to pick the first digit, 4 ways to pick the second digit, and 3 ways to pick the third.
Another way to think about this is there are ways to pick any three of the digits, and ways to rearrange them, so there are ways to pick and arrange 3 of the digits.
Then, we have to subtract cases where 0 is the first digit, because that would give us numbers such as 028 or 042 which are not three-digit natural numbers.
When 0 is the first digit, there are 4 possibilities for the second digit and 3 possibilities for the third digit, so there are ways for 0 to be the first digit.
These are the cases we don't want, so we subtract them from the total cases we can arrange any three of the digits. This gives us
total ways.
Solution 2:
We split the possible arrangements into two different cases: when we use the digits without 0 and when we do use the digit 0.
Case 1: we use 0:
We first pick the three numbers we use: one of them has to be 0, so there are ways to pick the other two numbers.
Then, since 0 cannot be the first digit, there are 2 ways to pick the first digit. We used a digit for the first number, so there are 2 more numbers we can choose from, including 0, for the second digit. There is only one possibility for the last digit, so we have ways to pick the order.
So, multiplying this by the number of ways there are to pick the numbers, there are ways for this case.
Case 2: we don't use 0:
If we don't use 0, there are 4 digits we can choose from: 2, 4, 6, and 8. There are 4 ways to choose the first digit, 3 ways to chose the second digit, and 2 ways for the third.
So, we have ways for this case.
Combining the cases, there are total possible ways to form a three-digit natural number using only the even digits.