Solution:
To find the number of seven-digit integers with a sum of digits equal to 11 using the digits 1, 2, and 3, we can break down the problem into smaller steps.

Step 1: Determine the possible positions for the digit 1.
Since the sum of the digits is 11, we know that at least one digit must be 1. Let's consider the possible positions for the digit 1 in a seven-digit number:

- The first digit cannot be 1 since it should be non-zero.
- The second digit can be 1.
- The third digit can be 1.
- The fourth digit can be 1.
- The fifth digit can be 1.
- The sixth digit can be 1.
- The seventh digit can be 1.

Therefore, we have seven possible positions for the digit 1.

Step 2: Determine the possible positions for the digit 2 and 3.
After placing the digit 1 in one of the seven positions, we need to determine the positions for the remaining digits 2 and 3. Since the sum of the digits is 11, the remaining digits must sum up to 10. Let's consider the possible positions for the digit 2:

- If the second digit is 2, then the remaining digits must sum up to 8.
- If the third digit is 2, then the remaining digits must sum up to 8.
- If the fourth digit is 2, then the remaining digits must sum up to 8.
- If the fifth digit is 2, then the remaining digits must sum up to 8.
- If the sixth digit is 2, then the remaining digits must sum up to 8.
- If the seventh digit is 2, then the remaining digits must sum up to 8.

Similarly, we can determine the possible positions for the digit 3.

Step 3: Count the number of possible combinations.
Now that we know the possible positions for each digit, we can count the number of combinations. For each position of digit 1, there are six possible positions for digit 2 and five possible positions for digit 3. Therefore, the total number of combinations is 7 * 6 * 5 = 210.

Step 4: Remove invalid combinations.
Not all combinations will result in a seven-digit number with a sum of digits equal to 11. We need to remove the invalid combinations.

- If the second digit is 1, then the remaining digits must sum up to 10. Since we only have the digits 2 and 3 remaining, the sum of the remaining digits cannot be 10. Therefore, we remove this combination.
- Similarly, we remove the combinations where the third, fourth, fifth, sixth, or seventh digit is 1.

After removing the invalid combinations, we are left with valid combinations.

Step 5: Count the number of valid combinations.
Count the number of valid combinations obtained from Step 4. In this case, there are 6 valid combinations. Therefore, the total number of seven-digit integers with a sum of digits equal to 11 is 6.

Therefore, the correct answer is 6.