Count All Valid Pickup and Delivery Options - Solution & Explanation

This approach leverages recursion with memoization to calculate the number of valid sequences. The main idea is to determine how many ways we can arrange pickups and deliveries by handling one order at a time.

This Python function uses a recursive helper function dp(p, d) that tracks how many pickups and deliveries are remaining. If both pickups and deliveries are zero, it returns 1 as a valid sequence is formed. Otherwise, it calculates the possible sequences either by picking up an order or delivering an order, avoiding violating the sequence rule.

This approach uses a combinatorial method by calculating factorial operations iteratively to compute the number of valid sequences. It simplifies the problem by deriving a general formula.

This C++ solution iteratively applies a combinatorial formula that derives from possible positioning of deliveries after pickups. It computes using factorial increases corresponding to the number of current orders.