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

Sure, let's solve this step by step:

1. First, we need to identify the integers between 700 and 1000 where the sum of the digits equals 10.

2. The integers between 700 and 1000 are three-digit numbers. So, we need to find the combinations of three digits that add up to 10.

3. The first digit of the numbers we are looking for is either 7, 8, or 9 because the numbers are between 700 and 1000.

4. If the first digit is 7, the other two digits can be (1,2), (2,1), (3,0), or (0,3). That gives us 4 possibilities.

5. If the first digit is 8, the other two digits can be (1,1), (2,0), or (0,2). That gives us 3 possibilities.

6. If the first digit is 9, the other two digits can only be (1,0) or (0,1). That gives us 2 possibilities.

7. Adding up all the possibilities, we get 4 + 3 + 2 = 9.

So, there are 9 integers between 700 and 1000 where the sum of the digits equals 10.