Leetcode #2400: Number of Ways to Reach a Position After Exactly k Steps
In this guide, we solve Leetcode #2400 Number of Ways to Reach a Position After Exactly k Steps in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
You are given two positive integers startPos and endPos. Initially, you are standing at position startPos on an infinite number line.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Math, Dynamic Programming, Combinatorics
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: startPos = 1, endPos = 2, k = 3
Output: 3
Explanation: We can reach position 2 from 1 in exactly 3 steps in three ways:
- 1 -> 2 -> 3 -> 2.
- 1 -> 2 -> 1 -> 2.
- 1 -> 0 -> 1 -> 2.
It can be proven that no other way is possible, so we return 3.
Python Solution
class Solution:
def numberOfWays(self, startPos: int, endPos: int, k: int) -> int:
def dfs(i: int, j: int) -> int:
if i > j or j < 0:
return 0
if j == 0:
return 1 if i == 0 else 0
return (dfs(i + 1, j - 1) + dfs(abs(i - 1), j - 1)) % mod
mod = 10**9 + 7
return dfs(abs(startPos - endPos), k)
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.