The 0/1 knapsack recurrence (max value using first i items, capacity c) is

Each item has two options: skip (value unchanged) or take (gain v_i, capacity drops by w_i). The recurrence is the max of the two: K[i][c] = max(K[i-1][c], v_i + K[i-1][c - w_i]), only taking when w_i ≤ c. Always-taking ignores capacity overflow; the other forms confuse table dimensions.