The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is __________.

Solution

Maths sir · 4 steps

Turn the number into three digit boxes, use stars and bars, then remove the cases where a digit goes above 9.

1. Set up digits. Write the number as abc with a + b + c = 14, where 1 ≤ a ≤ 9 and 0 ≤ b, c ≤ 9. Put a' = a - 1, so a' + b + c = 13 with a' ≤ 8 and b, c ≤ 9.
2. Count without limits. Non-negative solutions of a' + b + c = 13 number C(15, 2) = 105.
3. Remove bad cases. If a' ≥ 9, the rest sum to 4, giving C(6, 2) = 15. If b ≥ 10, the rest sum to 3, giving C(5, 2) = 10, and the same 10 for c ≥ 10. No two of these can happen together, because that would need a sum of at least 19.

   105 - (15 + 10 + 10) = 70

4. So. There are 70 such integers.

The common slip: 105. Skipping the subtraction for digits above 9 gives 105, not 70. That count includes impossible digits like 13 or 10.

The idea to keep. Digit-sum counting is stars and bars with caps. Shift the first digit so it starts at 0, count freely, then subtract every case where a digit breaks its upper limit.

Answer from the NTA final answer key. Solution by PrepWiser.