Answer:
8019 possible 5 digit numbers
Explanation:
Available number to pick from are a total of 10 which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
>> Now, let's consider when the first digit is 6 since we are told that 6 can appear only once.
For the first digit, it's only 1 way.
For the 2nd, 3rd, and 4th digit, we have 9 digits to pick from. Which is 9 ways each.
While for the 5th digit, it can't be 6 or 7. And it has to be even and so, we can pick only 0, 2, 4. Thus, we have 3 ways to choose.
So total number of ways here = 1 × 9 × 9 × 9 × 3 = 3 × 9³ ways
>> Now, let's consider when the 2nd digit is 6 since we are told that 6 can appear only once.
For the 1st digit, since 6 is not possible, and also 0 is not possible because a 5 digit number can't start with zero as it would no longer be a 5 digit number.
Thus, there are 8 ways to choose here.
For the second, third and fourth digits, there will be 9 ways each to chops.
While for the fifth digit, there will be 3 ways to choose.
Total number of ways here = 8 × 1 × 9 × 9 × 3 = 24 × 9² ways
>> When the third digit is 6, number of ways to choose will be same as above = 24 × 9² ways
>> When the fourth digit is 6, number of ways to choose will be same as above = 24 × 9² ways
We can't say none of the digits is 6 because we are told it appears exactly once.
Thus, total possible 5 digit numbers are;
(3 × 9³) + 3(24 × 9²) = 8019 possible 5 digit numbers