Using the numbers $0,1,2,3,4,5,6,7$ (repetition allowed) how many odd numbers can be created which will be less than $10000$? 
Using the numbers $0,1,2,3,4,5,6,7$ (repetition allowed) how many odd numbers can be created which will be less than $10000$?

I have tried to solve this in the following way:
Numbers which will be less than $10000$ must be one digit, two digits, three digits, or four digits (like $1,13,123,1235$ etc).
Therefore, one-digit odd numbers $=4$.
Two-digit odd numbers possible $=32-4=28$ (as there can be one zero)
Three-digit odd numbers possible $=256-28-4=224$ (again there can be two zeros)
Four-digit odd numbers possible $=2048-228-28-4=1788$ (as there can be three zeros)
Therefore, total possible numbers will be $=4+28+224+1788=2044$
But the book says the answer is $2048$.
Could you please solve this?

If you have Mathematica here is a way to check your answer. As it has been pointed out the mistake is in the number of four-digit odd numbers. 
a[n_] := 
  If[ContainsNone[IntegerDigits[n], {8}] && 
    ContainsNone[IntegerDigits[n], {9}] && OddQ[n], 1, 0];
Total[a /@ Table[i, {i, 1000, 9999}]]
