Respuesta :
Answer:
48
Step-by-step explanation:
Solution 1:
There are 5 even digits: 0, 2, 4, 6, and 8, and [tex]5\cdot 4\cdot 3=60[/tex] 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 [tex]5\choose 3[/tex] ways to pick any three of the digits, and [tex]3![/tex] ways to rearrange them, so there are [tex]{5\choose 3 }\cdot 3!=60[/tex] 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 [tex]4\cdot3=12[/tex] 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
[tex]60-12=\boxed{48}[/tex]
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 [tex]{4\choose 2}=6[/tex] 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 [tex]2\cdot2\cdot1=4[/tex] ways to pick the order.
So, multiplying this by the number of ways there are to pick the numbers, there are [tex]6\cdot4=24[/tex] 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 [tex]4\cdot3\cdot2=24[/tex] ways for this case.
Combining the cases, there are [tex]24+24=\boxed{48}[/tex] total possible ways to form a three-digit natural number using only the even digits.
