There are $2, 250.$ The reason is as follows. Let $\text{A}$ be the set of four-digit numbers that are odd. Let $\text{B}$ be the set of odd four-digit numbers that have the property that the sum of their digits is even.
The set we are interested in is $\text{A - B.}$ Now $\text{B} \subseteq \text{A}$ and thus
$|\text{A - B}| = |\text{A}| - |\text{B}|$
$= 4500 - 2250$
$= 2250,$
where $|\text{A}| = 4500$ and $|\text{B}| = 2250$
Why $|\text{A}| = 4500?$
There are $9 \cdot 10^2 \cdot 5 = 4500.$ The recipe is as follows:
- Choose the first digit (which can’t be $0) - 9$ choices
- Choose the second digit – $10$ choices
- Choose the third digit – $10$ choices
- Choose the fourth digit – $5$ choices (the last digit must be odd – only $5$ choices)
Why $|\text{B}| = 2250?$
There are $5 \cdot 9 \cdot 10 \cdot 5 = 2, 250.$ The recipe is as follows.
- Choose the last digit (which must be an odd digit) – $5$ choices
- Choose the first digit (which can’t be zero) – $9$ choices
- Choose the second digit – $10$ choices
- Choose the third digit (two cases) – $5$ choices
- Case i: If the sum of the other three digits is even, the third digit must be an even digit.
- Case ii: If the sum of the other three digits is odd, the third digit must be an odd digit