Question

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

Answer 1

Correct Answer: 70

To determine the count of integers between 100 and 1000 inclusive, where the sum of their digits equals 14, we represent such a number as a three-digit integer \(abc\). The constraints are as follows:

• The number must be within the range: \(100 \leq 100a + 10b + c<1000\)
• The sum of the digits must be 14: \(a + b + c = 14\)
• The first digit \(a\) must be between 1 and 9, inclusive: \(1 \leq a \leq 9\) (to ensure it's a three-digit number)

From these, we can establish the following bounds for each digit:

• \(1 \leq a \leq 9\)
• \(0 \leq b \leq 9\)
• \(0 \leq c \leq 9\)

For each possible value of \(a\) from 1 to 9, we enumerate the pairs of \(b\) and \(c\) that satisfy \(b + c = 14 - a\), keeping in mind that \(b\) and \(c\) must be single digits (0-9).

• When \(a = 1\), \(b + c = 13\). Valid pairs (b, c): (4,9), (5,8), (6,7), (7,6), (8,5), (9,4) - 6 combinations.
• When \(a = 2\), \(b + c = 12\). Valid pairs (b, c): (3,9), (4,8), (5,7), (6,6), (7,5), (8,4), (9,3) - 7 combinations.
• When \(a = 3\), \(b + c = 11\). Valid pairs (b, c): (2,9), (3,8), (4,7), (5,6), (6,5), (7,4), (8,3), (9,2) - 8 combinations.
• When \(a = 4\), \(b + c = 10\). Valid pairs (b, c): (1,9), (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2), (9,1) - 9 combinations.
• When \(a = 5\), \(b + c = 9\). Valid pairs (b, c): (0,9), (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1), (9,0) - 10 combinations.
• When \(a = 6\), \(b + c = 8\). Valid pairs (b, c): (0,8), (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1), (8,0) - 9 combinations.
• When \(a = 7\), \(b + c = 7\). Valid pairs (b, c): (0,7), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (7,0) - 8 combinations.
• When \(a = 8\), \(b + c = 6\). Valid pairs (b, c): (0,6), (1,5), (2,4), (3,3), (4,2), (5,1), (6,0) - 7 combinations.
• When \(a = 9\), \(b + c = 5\). Valid pairs (b, c): (0,5), (1,4), (2,3), (3,2), (4,1), (5,0) - 6 combinations.

Summing the counts for each value of \(a\): 6 + 7 + 8 + 9 + 10 + 9 + 8 + 7 + 6 = 70.

The total count of such integers is 70. This result aligns with the provided range of 70.70, indicating correctness.