Final answer:
To find the number of integers between 1 and 9,999 with digits summing to 10, we use combinatorics, placing 10 items into four boxes with restrictions based on digit limits. Counts must exclude leading zeros to ensure no number falls outside the specified interval.
Step-by-step explanation:
The question at hand is how many integers in the interval [1, 9 999] have digits whose sum equals 10. We must consider the different ways to sum digits to reach 10 without exceeding four digits. For example, the number 1900 has digits that sum to 10, as does the number 28.
We approach this problem by systematically enumerating possibilities using combinatorics. We can visualize this as placing 10 indistinguishable items (representing the sum of the digits) into 4 distinguishable boxes (representing the four digit places), with the condition that the first box (the thousands place) can't contain more than 9 items and the rest can't contain more than 9. We must account for leading zeros, which mean some configurations actually represent three, two, or one-digit numbers.
By considering all valid combinations and careful counting, we could determine the exact quantity of valid numbers, ensuring we stay between 1 and 9,999 and keeping the digits' sum at 10.