Maximize Revenue from Falling Color Inventories

### Problem `inventory[i]` is the stock of color i. Selling one ball of that color earns its current stock count, then decreases that count by one. Sell exactly `orders` balls, choosing the colors and sale order freely. Return the largest total revenue modulo `10^9 + 7`. ```cpp class Solution { public: int maxProfit(vector<int>& inventory, int orders); }; ``` ### Examples **Example 1:** ```text inventory = [1,4], orders = 3 Output: 9 Explanation: Sell from the second color at values 4, 3, and 2. ``` **Example 2:** ```text inventory = [3,3], orders = 4 Output: 10 Explanation: The four largest available values are 3, 3, 2, and 2. ``` ### Constraints - `1 <= inventory.length <= 10^5`. - `1 <= inventory[i] <= 10^9`. - `1 <= orders <= min(sum(inventory), 10^9)`. ### Clarifying Questions The value is measured before the sale. Only the maximum revenue is returned; several color orders can attain that same value. ```hint Inspect the structure When is a ball's value measured, and what does selling exactly `orders` require? ``` ### Reasoning checkpoints - Account for changing sale values and sell exactly the requested number of balls. - Prove that the revenue is maximal and handle the stated stock and order bounds without arithmetic overflow. ### Optional extensions How could sorting inventory heights replace the cutoff binary search?