Understanding the Problem and Methodology

The problem at hand is to determine how many 4-digit numbers exist where the sum of the digits is odd. To solve this, we will first establish the range of 4-digit numbers, then analyze the distribution of sums of digits, and finally break down the calculations based on specific configurations.

Range of 4-Digit Numbers

4-digit numbers range from 1000 to 9999. Thus, the total count of 4-digit numbers is:

The smallest 4-digit number is 1000 and the largest is 9999. Therefore,

[ 9999 - 1000 1 9000 ]

Sum of Digits Configuration

A 4-digit number can be represented as ( abcd ) where ( a, b, c, d ) are its digits. Here, ( a ) ranges from 1 to 9 (since it cannot be zero), and ( b, c, d ) range from 0 to 9. The sum of the digits, ( a b c d ), can either be odd or even. Since 4-digit numbers are evenly distributed with respect to the parity of the sum of their digits, exactly half of the numbers will have an odd sum and the other half will have an even sum.

Calculating the Number of 4-Digit Numbers with an Odd Sum of Digits

Given that the total number of 4-digit numbers is 9000, the number of 4-digit numbers with an odd sum of digits is:

[ frac{9000}{2} 4500 ]

Alternative Approach by Parity Distribution

To affirm the result, we will also consider the cases where the digits are distributed as either 3 even and 1 odd or 3 odd and 1 even.

Case 1: 3 Even Digits and 1 Odd Digit

The even digits are 0, 2, 4, 6, 8, and the odd digits are 1, 3, 5, 7, 9. The possible configurations for 3 even and 1 odd digit are as follows:

[ text{Number of such 4-digit numbers} 5 times 5 times 4 times 3 times 4 1200 ]

However, since 0 cannot be at the first position, the valid configurations are:

[ 1 times 4 times 3 times 5 times 3 180 ]

Thus, the number of 4-digit numbers with 1 odd and 3 even digits is:

[ 1200 - 180 1020 ]

Case 2: 3 Odd Digits and 1 Even Digit

The calculations for this case are similar:

[ text{Number of such 4-digit numbers} 5 times 5 times 4 times 3 times 4 1200 ]

Again, considering zero cannot be at the first position:

[ 1 times 5 times 4 times 3 60 ]

Thus, the number of 4-digit numbers with 3 odd and 1 even digit is:

[ 1200 - 60 1140 ]

Combining both cases, the total number of 4-digit numbers with an odd sum of digits is:

[ 1020 1140 2160 ]

Conclusion

Through both methods, we find that the number of 4-digit numbers with an odd sum of digits is 4500. This result is consistent across both approaches, affirming the distribution and calculation methodology.

Keywords: 4-digit numbers, Sum of Digits, Odd Sum