To find the number of three-digit positive integers that contain both even and odd digits, we can follow these steps:
### Step 1: Calculate the total number of three-digit numbers.
Three-digit numbers range from 100 to 999.
- The first digit (hundreds place) can be any digit from 1 to 9 (9 options).
- The second digit (tens place) can be any digit from 0 to 9 (10 options).
- The third digit (units place) can also be any digit from 0 to 9 (10 options).
Thus, the total number of three-digit numbers is calculated as:
\[
\text{Total three-digit numbers} = 9 \times 10 \times 10 = 900
\]
### Step 2: Calculate the number of three-digit numbers with only even digits.
The even digits from 0 to 9 are 0, 2, 4, 6, and 8.
- The first digit (hundreds place) can be any even digit except 0 (4 options: 2, 4, 6, 8).
- The second digit (tens place) can be any even digit (5 options: 0, 2, 4, 6, 8).
- The third digit (units place) can also be any even digit (5 options: 0, 2, 4, 6, 8).
Thus, the total number of three-digit numbers with only even digits is calculated as:
\[
\text{Total even-digit numbers} = 4 \times 5 \times 5 = 100
\]
### Step 3: Calculate the number of three-digit numbers with only odd digits.
The odd digits from 0 to 9 are 1, 3, 5, 7, and 9.
- The first digit (hundreds place) can be any odd digit (5 options: 1, 3, 5, 7, 9).
- The second digit (tens place) can also be any odd digit (5 options: 1, 3, 5, 7, 9).
- The third digit (units place) can also be any odd digit (5 options: 1, 3, 5, 7, 9).
Thus, the total number of three-digit numbers with only odd digits is calculated as:
\[
\text{Total odd-digit numbers} = 5 \times 5 \times 5 = 125
\]
### Step 4: Calculate the number of three-digit numbers with both even and odd digits.
To find the count of three-digit numbers that have both even and odd digits, we can subtract the numbers that have only even digits and only odd digits from the total count of three-digit numbers:
\[
\text{Numbers with both even and odd digits} = \text{Total three-digit numbers} - \text{Total even-digit numbers} - \text{Total odd-digit numbers}
\]
Substituting the values we calculated:
\[
\text{Numbers with both even and odd digits} = 900 - 100 - 125 = 675
\]
### Final Answer
The number of three-digit positive integers that contains both even and odd digits is **675**.
---
