How many different integers can be expressed as the sum of three distinct…

If every element of a set is congruent to a fixed remainder r modulo m, then the sum of any k of them is congruent to k·r (mod m) — so any two valid k-element sums differ by a multiple of m. That alone only shows the sums lie on an evenly spaced grid with common difference m; it does not by itself guarantee that every grid value between the smallest and largest sum is actually attained. That gap-free property needs a separate argument: when the k elements are drawn from a block of n index-consecutive terms, a k-subset that is not already the top k indices can always be advanced to a 'one-larger' k-subset by moving a single chosen index to its unused successor, which increases the sum by exactly 1 — so repeating this step from the minimum reaches the maximum without skipping any grid value, confirming the count of distinct sums is exactly (maximum − minimum)/m + 1.

1. Every element of {3, 10, 17, 24, 31, 38, 45, 52} has the form 3 + 7i for i = 0, 1, …, 7 (index 0 through index 7), so each element is ≡ 3 (mod 7).

2. A sum of any three distinct elements is therefore ≡ 3 + 3 + 3 = 9 ≡ 2 (mod 7); every valid sum shares this residue, so any two valid sums differ by a multiple of 7.

3. The minimum sum uses the three smallest elements (indices 0, 1, 2): 3 + 10 + 17 = 30.

4. The maximum sum uses the three largest elements (indices 5, 6, 7): 38 + 45 + 52 = 135.

5. By the exchange argument above, every multiple-of-7 step between 30 and 135 is actually attained, so the count is (135 − 30)/7 + 1 = 105/7 + 1 = 15 + 1 = 16.

Cross-check in terms of indices: three distinct indices from {0, 1, …, 7} have a sum ranging from the minimum 0 + 1 + 2 = 3 to the maximum 5 + 6 + 7 = 18. Whenever the current index-sum is below the maximum, the three chosen indices cannot all already have their successor also chosen (that would force them to be exactly {5, 6, 7}), so some chosen index x has an unused successor x + 1; replacing x by x + 1 increases the sum by exactly 1. Repeating this from the minimum reaches the maximum one step at a time, so every integer from 3 to 18 occurs — 16 values, matching the count above.

Hence 16 different integers can be expressed as the sum of three distinct numbers from the given set.