Question

If O represents the number of integers between 10000 and 100000 all whose digits are odd, and E represents all the number of integers between 10000 and 100000 all of whose digits are even, what is the value of O-E?

Answer 1

All numbers within the range have 5 digits. For any one digit, you have 5 choices (1,3,5,7, or 9), so \(O=5^5\).

You can determine \(E\) similarly, but keep in mind that the first digit can't be \(0\).

Answer 2

Oh, I get it now. I thought it meant the amount of odd numbers between 10000 and 100000, which would be a lot more

Answer 3

So for E, even numbers, you would have 4 choices (2,4,6,8)

Answer 4

Right, if we're only counting odd numbers we only need to check the last digit.

The first digit of every number in \(E\) has 4 choices, yes (2,4,6, or 8), but every digit after that can also be a \(0\), so those have 5 choices.

Answer 5

And since the other digits can be 0, that should be 5^4

Answer 6

so in total

O=5^5
E=4x5x5x5x5

Answer 7

Correct

Answer 8

and from there it's just basic subtraction

Answer 9

thanks man

Answer 10

No problem. You can avoid actually computing the powers, though, and instead make a slight manipulation:
\[5^5-4\times5^4=5^4(5-4)=5^4\]