Count Numbers with Unique Digits

LeetCode 357 | Difficulty: Medium

Medium

Problem Description

Given an integer n, return the count of all numbers with unique digits, x, where 0 <= x < 10^n.

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`

Topics: Math, Dynamic Programming, Backtracking

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

Backtracking

Explore all candidates by building solutions incrementally. At each step, choose an option, explore further, then unchoose (backtrack) to try the next option. Prune branches that can't lead to valid solutions.

When to use

Generate all combinations/permutations, or find solutions that satisfy constraints.

Mathematical

Look for mathematical patterns or formulas. Consider: modular arithmetic, GCD/LCM, prime factorization, combinatorics, or geometric properties.

When to use

Problems with clear mathematical structure, counting, number properties.

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Solutions

Solution 1: C# (Best: 40 ms)

Metric	Value
Runtime	40 ms
Memory	25.3 MB
Date	2022-01-17

Solution

public class Solution {
    public int CountNumbersWithUniqueDigits(int n) {
        if(n==0) return 1;
        int count = 9, total = 10;
        if(n>10) n = 10;
        for(int i=1;i<n;i++)
        {
            count = count * (10-i);
            total += count;
        }
        
        return total;
    }
}

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Complexity Analysis

Approach	Time	Space
Dynamic Programming	$O(n)$	$O(n)$
Backtracking	$O(n! or 2^n)$	$O(n)$

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Interview Tips

Key Points

• Discuss the brute force approach first, then optimize. Explain your thought process.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.
• Identify pruning conditions early to avoid exploring invalid branches.
• LeetCode provides 5 hint(s) for this problem — try solving without them first.

💡 Hints

Hint 1: A direct way is to use the backtracking approach.

Hint 2: Backtracking should contains three states which are (the current number, number of steps to get that number and a bitmask which represent which number is marked as visited so far in the current number). Start with state (0,0,0) and count all valid number till we reach number of steps equals to 10^n.

Hint 3: This problem can also be solved using a dynamic programming approach and some knowledge of combinatorics.

Hint 4: Let f(k) = count of numbers with unique digits with length equals k.

Hint 5: f(1) = 10, ..., f(k) = 9 9 8 * ... (9 - k + 2) [The first factor is 9 because a number cannot start with 0].