357. Count Numbers with Unique Digits

Given a non-negative integer n, count all numbers with unique digits, x, where 0 ≤ x < 10n.

Example:

Given n = 2, return 91. (The answer should be the total numbers in the range of 0 ≤ x < 100, excluding [11,22,33,44,55,66,77,88,99])

Hint:

A direct way is to use the backtracking approach.

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 10n.

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

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

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

The Unique digits here means no-duplicates digits for this number.

dp, if n:

• 0, result is 1, f(0)
• 1, 10, which is 9 + f(0);
• 2, 91, which is 9*9 + f(1), first number gets 9 choice(no zero), next is still 9(with zero),
• 3, 9*9*8 + f(2), each bit will get a choice of 9 - (i-2), i is current bit, -2 means excludes first two bits.

public class Solution {
    public int countNumbersWithUniqueDigits(int n) {
        int res = 1;
        for(int i = 1; i<=n;i++){
            int tmp = 9;
            // j>0 here is redundant, the number with no duplicates should < 10-bit, otherwise there must be a duplicates.
            for(int j =9; j>0 && j>= 9-i+2; j--)
                tmp *= j;

            res += tmp;
        }

        return res;
    }
}