Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?

The problem asks us to find the count of integers between 700 and 1000 (inclusive) where the sum of the digits is 10.

We categorize the numbers based on their first digit:

1. Numbers starting with 7 (7xx): Let the number be 7bc. The sum of digits is 7 + b + c = 10, which means b + c = 3. Possible pairs for (b, c) are:

   • (0, 3) -> 703
   • (1, 2) -> 712
   • (2, 1) -> 721
   • (3, 0) -> 730 There are 4 such numbers.

2. Numbers starting with 8 (8xx): Let the number be 8bc. The sum of digits is 8 + b + c = 10, which means b + c = 2. Possible pairs for (b, c) are:

   • (0, 2) -> 802
   • (1, 1) -> 811
   • (2, 0) -> 820 There are 3 such numbers.

3. Numbers starting with 9 (9xx): Let the number be 9bc. The sum of digits is 9 + b + c = 10, which means b + c = 1. Possible pairs for (b, c) are:

   • (0, 1) -> 901
   • (1, 0) -> 910 There are 2 such numbers.

4. The number 1000: The sum of digits for 1000 is 1 + 0 + 0 + 0 = 1, which is not 10. So, 1000 is not included.

Total count of integers = 4 (from 7xx) + 3 (from 8xx) + 2 (from 9xx) = 9.

Therefore, the correct answer is D) 9.