**Python Implementation:** ```python def knapsack(values, w…

**Python Implementation:**
```python
def knapsack(values, weights, W):
n = len(values)
# Initialize the dp table with 0
dp = [[0 for _ in range(W + 1)] for _ in range(n + 1)]
for i in range(1, n + 1):
for w in range(W + 1):
if weights[i-1] <= w:
dp[i][w] = max(dp[i-1][w], dp[i-1][w-weights[i-1]] + values[i-1])
else:
dp[i][w] = dp[i-1][w]
# The maximum value that can be carried is in the bottom-right corner of the matrix
return dp[n][W]
# Example usage:
values = [60, 100, 120]
weights = [10, 20, 30]
W = 50
max_value = knapsack(values, weights, W)
print("The maximum value that can be put in the knapsack is:", max_value)
```
**Explanation:**
- We iterate over each item and sub-capacity (0 to W).
- For each item/sub-capacity pair, we determine if the item can be included. If it can, we compare the value of including it versus not including it.
- We use the precomputed results stored in the `dp` table to ensure efficiency, avoiding recalculation and yielding a time complexity of O(nW), which is much more efficient than the naive approach.