Leetcode #357: Count Numbers with Unique Digits

In this guide, we solve Leetcode #357 Count Numbers with Unique Digits 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

Given an integer n, return the count of all numbers with unique digits, x, where 0 <= x < 10n. Example 1: Input: n = 2 Output: 91 Explanation: The answer should be the total numbers in the range of 0 ≤ x < 100, excluding 11,22,33,44,55,66,77,88,99 Example 2: Input: n = 0 Output: 1 Constraints: 0 <= n <= 8

Quick Facts

• Difficulty: Medium
• Premium: No
• Tags: Math, Dynamic Programming, Backtracking

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

Python Solution

Complexity

The time complexity is $O(n \times 2^D \times D)$, and the space complexity is $O(n \times 2^D)$. The space complexity is $O(n \times 2^D)$.

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.