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

The correct option is 9.

We are asked to find how many integers between 700 and 1000 (inclusive) have a digit sum equal to 10.

Let any three-digit integer in this range be represented as abc, where a is the hundreds digit, b is the tens digit, and c is the units digit.

Since the range is from 700 to 1000:

• The integer 1000 has a digit sum of 1 + 0 + 0 + 0 = 1, which is not equal to 10.

• Therefore, all possible integers must be three-digit numbers starting with 7, 8, or 9.

The condition for the sum of the digits to be 10 is:

$a+b+c=10$

Let us analyze the possible cases based on the hundreds digit a:

Case 1: Hundreds digit a = 7

$7+b+c=10\Rightarrow b+c=3$

The valid non-negative single-digit pairs (b, c) that add up to 3 are:

• (0, 3) ⇒ 703

• (1, 2) ⇒ 712

• (2, 1) ⇒ 721

• (3, 0) ⇒ 730

This gives 4 such integers.

Case 2: Hundreds digit a = 8

$8+b+c=10\Rightarrow b+c=2$

The valid non-negative single-digit pairs (b, c) that add up to 2 are:

• (0, 2) ⇒ 802

• (1, 1) ⇒ 811

• (2, 0) ⇒ 820

This gives 3 such integers.

Case 3: Hundreds digit a = 9

$9+b+c=10\Rightarrow b+c=1$

The valid non-negative single-digit pairs (b, c) that add up to 1 are:

• (0, 1) ⇒ 901

• (1, 0) ⇒ 910

This gives 2 such integers.

Total Count:

Adding the number of valid integers from all cases:

$\text{Total}=4+3+2=9$

Thus, there are 9 integers from 700 to 1000 in which the sum of the digits is 10.