Abstract / Summary

Count the number of ways to arrange pickup and delivery orders so each delivery happens after its matching pickup.

Note

The full problem description, including helpful examples, can be found on LeetCode.

This problem is ranked as

and has an acceptance rate of about 65% (100k+ submissions).

Typical patterns to solve this problem:

• Combinatorics
• Dynamic Programming
• Math

Every single line of code was written by me, 100% from scratch, between August 2023 and August 2024.
However, two years later I set up this website and used various AI tools to generate most of the walkthrough / explanations below.

See this dedicated page for more details.

Overview

The provided implementation defines a single class Solution with a method countOrders. The method computes the number of valid pickup-delivery sequences for n orders, a combinatorial quantity known to follow the sequence A000680. The calculation is performed in a single arithmetic expression that combines factorial evaluation, integer division, exponentiation, and modular reduction.

Key Steps

The core of the algorithm resides in the return statement:

return (factorial(2 * n) // 2**n) % (10**9 + 7)

The expression can be broken down into three conceptual stages. First, factorial(2 * n) generates the factorial of twice the input size, representing the total number of permutations of 2n events (pickups and deliveries) without any constraints. Second, the integer division // 2**n removes the over-counting caused by the requirement that each delivery must occur after its corresponding pickup; this division corresponds to the factor 2^n in the closed-form formula. Finally, the modulo operation % (10**9 + 7) ensures the result fits within the typical 32-bit integer range required by many programming challenges.

The surrounding class structure is minimal:

class Solution:
    def countOrders(self, n: int) -> int:
        # implementation as shown above

No additional helper functions or loops are introduced, keeping the control flow straightforward.

Edge Cases Handled

When n equals zero, the expression evaluates as follows: factorial(0) yields 1, 2**0 also yields 1, and the division 1 // 1 results in 1. The final modulo operation leaves the value unchanged, correctly returning 1, which matches the combinatorial interpretation that there is exactly one way to arrange zero orders.

For very large values of n, the use of Python’s built-in factorial function leverages an implementation written in C, capable of handling arbitrarily large integers efficiently. The division by 2**n reduces the magnitude of the intermediate result before the modulo is applied, preventing the final reduction step from operating on an excessively large number. This ordering of operations ensures that the method remains correct even when n approaches the limits of typical problem constraints.

Performance Optimizations

The algorithm benefits from two notable optimizations. The first is the reliance on the standard library’s factorial function, which is highly optimized and executes in compiled code rather than pure Python loops. This choice eliminates the overhead of manually iterating to compute the factorial.

The second optimization is the placement of the integer division before the modulo operation. By dividing the factorial result by 2**n first, the size of the number subjected to the modulo reduction is significantly smaller. This ordering reduces the computational cost of the modular reduction, especially for large n where the factorial value would otherwise be astronomically large.

Together, these choices allow the method to compute the desired combinatorial count in a single, efficient expression without explicit iteration or recursion.

Imagine you have a row of empty boxes that need to be filled with packages. Each package needs to be picked up first and then delivered to someone. The code works by figuring out how many different ways those pickup and delivery tasks can be arranged for a certain number of orders. It looks at the total number of items, which is double the number of orders, and calculates all the possible ways to line them up. Then it divides that big pile of possibilities by a special number to make sure every pickup happens before its matching delivery. Because the answer can get incredibly large, the code wraps things up by using a giant number to keep the result neatly contained within a smaller range.

Several LLMs ranked my code quality on a scale from 1 (very poor) to 10 (outstanding):

Time: linear
As a rule of thumb, the maximum input size for one second execution time is

• factorial: up to 10 elements
• exponential: up to 20 elements
• polynomial: a few thousand elements
• loglinear: up to 100,000 elements
• linear: a few million elements
• logarithmic: a billion elements
• constant: practically unlimited

I wrote a basic crawler to find other solutions to that problem.
Obviously, I did that after I solved the problem myself.

In general, AlgoMonster and WalkCCC solutions are high-quality and always worth a look.
I haven't checked most of the other solutions, but they might be helpful as well.

And of course, the LeetCode forum is always a good place (or even the best) to look for alternative approaches.

(click on a column header to sort the table)

	author
author	solutions
www	AlgoMonster	C++		Python	Java			TS
	WalkCCC	C++		Python	Java
	leetcode.ca	C++		Python	Java	Go	Rust
	Progiez	C++		Python	Java
github	black-shadows	C++
	charles-wangkai				Java
	everthis		JS
	fsq	C++
	JoshCrozier		JS
	keineahnung2345	C++
	lkwq007			Python
	wisdompeak	C++
AlgoMonster	C++, Python, Java, TypeScript
WalkCCC	C++, Python, Java
leetcode.ca	C++, Go, Java, Python, Rust
Progiez	C++, Java, Python
black-shadows	C++, C++
charles-wangkai	Java
everthis	JavaScript
fsq	C++
JoshCrozier	JavaScript
keineahnung2345	C++
lkwq007	Python
wisdompeak	C++

There are a few extra pages giving you some insights into my setup:

• Tech Stack - my development environment and tools
• AI - how AI helped me create this website (NOT solving the problems)
• Progress - progression over time
• Quality - code quality