0/1 Knapsack - Walkthrough

Let's trace 0/1 Knapsack with items $[(w=2, v=3), (w=3, v=4), (w=4, v=5)]$ and capacity $W=5$. Build the DP table row by row.

After item 1: $dp[2]=3$.

After item 2: $dp[3]=4$, $dp[5]=7$ (take both).

After item 3: $dp[4]=5$, $dp[5]=\max(7, 5)=7$. The answer is $dp[5]=7$. We took items 1 and 2 (weights $2+3=5$, values $3+4=7$). Item 3 doesn't fit with either of them. Notice how each cell considers: skip this item (keep previous value) or take it (add value, subtract weight). This is the core of all knapsack variants.