# Count of N digit numbers with at least one digit as K

• Last Updated : 03 Aug, 2022

Given a number N and a digit K, The task is to count N digit numbers with at least one digit as K.

Examples:

Input: N = 3, K = 2
Output: 252
Explanation:
For one occurrence of 2 –
In a number of length 3, the following cases are possible:
=>When first digit is 2 and other two digits can have 9 values except ‘2’.
Thus 9*9 = 81 combination are possible.
=> When second digit is 2 and first digit can have 8 values from 1 to 9 except ‘2’
and the third digit can have 9 value from 0 to 9 except ‘2’.
Thus 8*9 = 72 valid combination.
=>When third digit is 2 the first digit can have 8 values from 1 to 9 except ‘2’
and the second digit can have 9 values from 0 to 9 except ‘2’ thus 8*9 = 72.
Hence total valid combination with one occurrence of 2 = 72 + 72 + 81 = 225.
For two occurrence of 2 –
First and second digit can be 2 and third digit can have 9 values from 0 to 9.
Second and third digit can have value 2 and first digit
can have  8 values from 1 to 9 except 2.
First and third digit can have values 2 and second digit
can have 9 values from 0 to 9 except 2.
Hence total valid combination with two occurrence of 2 = 9 + 8 + 9 = 26.
For all three digits to be 2 –
There can be only 1 combination.
Hence total possible numbers with at least one occurrence of 2 =  225 + 26 + 1 = 252.

Input: N = 9, K = 8
Output: 555626232

Approach: The problem can be solved based on the following mathematical idea:

Find the difference between count of unique N digit numbers possible and count of all unique N digit numbers with no occurrence of digit K.

Follow the steps mentioned below to implement this idea:

• Find the count of all N digits numbers = 9 x 10N-1, Leftmost place can be any digit from 1-9, other digits can have any value from between 0 and 9.
• Find the count of all N digits number with no occurrence of K = 8 x 9n-1, Leftmost place can be any digit from 1 to 9 except K and other digits can have any value between 0 to 9 except K.
• Total count of N digit numbers with at least one occurrence of
= Count of all N digits numbers –  Count of all N digit numbers with no occurrence of K.

Below is the implementation of the above approach:

## C++

 `// C++ Code to Implement the approach``// Function to find the total possible numbers``#include ``#include ``using` `namespace` `std;` `// Function to find the total possible numbers``void` `required_numbers(``int` `n, ``int` `k)``{``    ``int` `t, h, r;``  ` `    ``// Find all n digits numbers``    ``t = 9 * ``pow` `(10, (n - 1));``  ` `    ``// Find n digits number in which no k occurs``    ``h = 8 * ``pow` `(9, (n - 1));``  ` `    ``// Calculate the required value as``    ``// the difference of the above two values``    ``r = t - h;``    ``cout << r;``}` `// Driver code``int` `main()``{` `    ``int` `N, K;``    ``N = 3;``    ``K = 2;``  ` `    ``// Function call``    ``required_numbers(N, K);``    ``return` `0;``}` `// This code is contributed by ANKITKUMAR34.`

## Java

 `// Java Code to implement the approach``// Function to find the total possible numbers``import` `java.io.*;``import` `java.util.*;` `class` `GFG {``    ``// Function to find the total possible numbers``    ``public` `static` `void` `required_numbers(``int` `n, ``int` `k)``    ``{``        ``// Find all n digits numbers``        ``int` `t = ``9` `* (``int``)Math.pow(``10``, (n - ``1``));` `        ``// Find n digits number in which no k occurs``        ``int` `h = ``8` `* (``int``)Math.pow(``9``, (n - ``1``));` `        ``// Calculate the required value as``        ``// the difference of the above two values``        ``int` `r = t - h;``        ``System.out.print(r);``    ``}``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `N = ``3``;``        ``int` `K = ``2``;` `        ``// Function call``        ``required_numbers(N, K);``    ``}``}` `// This code is contributed by Rohit Pradhan`

## Python3

 `# Python Code to Implement the approach`  `# Function to find the total possible numbers``def` `required_numbers(n, k):` `    ``# Find all n digits numbers``    ``t ``=` `9` `*` `10` `*``*` `(n ``-` `1``)` `    ``# Find n digits number in which no k occurs``    ``h ``=` `8` `*` `9` `*``*` `(n ``-` `1``)` `    ``# Calculate the required value as``    ``# the difference of the above two values``    ``r ``=` `t ``-` `h``    ``return``(r)`  `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``N ``=` `3``    ``K ``=` `2` `    ``# Function call``    ``print``(required_numbers(N, K))`

## C#

 `// C# Code to implement the approach``// Function to find the total possible numbers` `using` `System;` `public` `class` `GFG {``    ` `    ``// Function to find the total possible numbers``    ``public` `static` `void` `required_numbers(``int` `n, ``int` `k)``    ``{``        ``// Find all n digits numbers``        ``int` `t = 9 * (``int``)Math.Pow(10, (n - 1));` `        ``// Find n digits number in which no k occurs``        ``int` `h = 8 * (``int``)Math.Pow(9, (n - 1));` `        ``// Calculate the required value as``        ``// the difference of the above two values``        ``int` `r = t - h;``        ` `        ``Console.WriteLine(r);``    ``}``    ` `    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``int` `N = 3;``        ``int` `K = 2;` `        ``// Function call``        ``required_numbers(N, K);``    ``}``}` `// This code is contributed by AnkThon`

## Javascript

 ``

Output

`252`

Time Complexity: O(1).
Auxiliary Space: O(1).

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