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

Correct Option: (b) 7

We need to find the number of integers from 700 to 1000 where the sum of the digits is equal to 10. Let the integer be represented as abc, where a is the hundreds digit, b is the tens digit, and c is the units digit. The range is 700 to 1000. Case 1: a = 7. Then 7 + b + c = 10 => b + c = 3. Possible pairs for (b, c) where 0 <= b <= 9 and 0 <= c <= 9: (0, 3), (1, 2), (2, 1), (3, 0). Total = 4 integers (703, 712, 721, 730). Case 2: a = 8. Then 8 + b + c = 10 => b + c = 2. Possible pairs for (b, c): (0, 2), (1, 1), (2, 0). Total = 3 integers (802, 811, 820). Case 3: a = 9. Then 9 + b + c = 10 => b + c = 1. Possible pairs for (b, c): (0, 1), (1, 0). Total = 2 integers (901, 910). Case 4: 1000 has digits 1 + 0 + 0 + 0 = 1 (not 10). Total integers = 4 + 3 + 2 = 9 integers? Wait, let's check: 4 + 3 + 2 = 9. Let's check the options: (a) 6, (b) 7, (c) 8, (d) 9. Wait, let's re-verify sum for a=7: b+c=3 gives (0,3)->703, (1,2)->712, (2,1)->721, (3,0)->730 (4 numbers). For a=8: b+c=2 gives (0,2)->802, (1,1)->811, (2,0)->820 (3 numbers). For a=9: b+c=1 gives (0,1)->901, (1,0)->910 (2 numbers). Total = 4 + 3 + 2 = 9. Wait, let's check if option (d) is 9. Options: (a) 6, (b) 7, (c) 8, (d) 9. Option (d) index is 3. Wait, let's check if 700 itself or 1000 has sum 10. 7+0+0=7. 1+0+0+0=1. Total is 9. Thus option (d) is correct, index 3.