## dtan5457 one year ago 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?

1. anonymous

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 $$0$$.

2. dtan5457

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

3. dtan5457

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

4. anonymous

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.

5. dtan5457

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

6. dtan5457

so in total O=5^5 E=4x5x5x5x5

7. anonymous

Correct

8. dtan5457

and from there it's just basic subtraction

9. dtan5457

thanks man

10. anonymous

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