Final answer:
The number of 5-digit numbers with an even digit sum is 90,000, while the number of 5-digit numbers with an odd digit sum is 6,250. Thus, x (number of 5-digit numbers with even digit sum) is not equal to y (number of 5-digit numbers with odd digit sum), and x + y = 90,000. Therefore, option B is correct: x + y = 90,000.
Step-by-step explanation:
Let's break down the problem and solve it step-by-step:
- To find the number of 5-digit numbers whose digits sum is even, we need to consider the options for each digit.
- For the first digit, it can be any number from 1 to 9 (as 0 is not allowed as the first digit of a 5-digit number).
- For the remaining four digits, each digit can be any number from 0 to 9.
- So, the total number of options is 9 (for the first digit) * 10 (for each of the remaining four digits).
- Therefore, the number of 5-digit numbers whose digits sum is even is 9 * 104 = 90,000.
- Similarly, to find the number of 5-digit numbers whose digits sum is odd, we follow the same steps, but with a slight modification.
- For the first digit, it can still be any number from 1 to 9.
- For the remaining four digits, each digit can still be any number from 0 to 9.
- However, we need to consider that the sum of digits should be odd, which means that the first digit should be odd if the rest of the digits are even, or the first digit should be even if the rest of the digits are odd.
- So, we have 5 options for the first digit (either 1, 3, 5, 7, or 9) and for each of the remaining four digits, each digit can be any even number from 0 to 8 (since 0 is not allowed as the first digit).
- Therefore, the total number of 5-digit numbers whose digits sum is odd is 5 * 54 = 6,250.
- Hence, x (the number of 5-digit numbers with even digit sum) is not equal to y (the number of 5-digit numbers with odd digit sum) and x + y = 90,000 + 6,250 = 96,250.
- Therefore, option B is correct: x + y = 90,000.