Count Numbers with N digits which consists of odd number of 0’s

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We are given a number N. The task is to find the count of numbers which have N digits and odd number of zeroes.

Note: The number can have preceding 0’s.

Examples:

Input : N = 2
Output : Count = 18

Input : N = 3
Output : Count = 244

Suppose a number with N digits which contains only single zero. So the digits in the number as zero can be filled in only 1 way and the rest of each of the positions can be filled in 9 different ways with numbers from 1 to 9. So count of all such numbers with N digits and only 1 zero = ^NC₁*(9^N-1).

Similarly, count of all such numbers with N digits and 3 zeroes = ^NC₃*(9^N-3).
and so on.

So, count of all such numbers with N digits and odd number of zeroes will be,

^NC₁*(9^N-1) + ^NC₃*(9^N-3) + ^NC₅*(9^N-5) +…….+ ^NC_K*(9^N-K)
Where, K is an odd number less than N.

The above equation can be written as,

(9^N)((^NC₁ * (1/9)) + (^NC₃ * (1/9^3)) + (^NC₅ * (1/9^5)) +…

The above equation can be represented as subtraction of two series, (9^N)*{(1+x)^N-(1-x)^N}/2, where x = 1/9
Which is equal to,

(10^{N - 8N)/2}

Below is the implementation of the above approach:

C++

// C++ program to count numbers with N digits
// which consists of odd number of 0's
#include <bits/stdc++.h>
using namespace std;
 
// Function to count Numbers with N digits
// which consists of odd number of 0's
int countNumbers(int N)
{
    return (pow(10, N) - pow(8, N)) / 2;
}
 
// Driver code
int main()
{
    int n = 5;
 
    cout << countNumbers(n) << endl;
 
    return 0;
}

Java

// Java program to count numbers
// with N digits which consists
// of odd number of 0's
import java.io.*;
 
class GFG {
 
    // Function to count Numbers
    // with N digits which consists
    // of odd number of 0's
    static int countNumbers(int N)
    {
        return (int)(Math.pow(10, N) 
                     - Math.pow(8, N)) / 2;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 5;
        System.out.println(countNumbers(n));
    }
}
 
// This code is contributed by Shashank

Python3

# Python 3 program to count numbers
# with N digits which consists of
# odd number of 0's
 
# Function to count Numbers with
# N digits which consists of odd
# number of 0's
 
 
def countNumbers(N):
 
    return (pow(10, N) - pow(8, N)) // 2
 
 
# Driver code
if __name__ == "__main__":
 
    n = 5
 
    print(countNumbers(n))
 
# This code is contributed
# by ChitraNayal

C#

// C# program to count numbers
// with N digits which consists
// of odd number of 0's
using System;
 
class GFG {
 
    // Function to count Numbers
    // with N digits which consists
    // of odd number of 0's
    static int countNumbers(int N)
    {
        return (int)(Math.Pow(10, N) 
                     - Math.Pow(8, N)) / 2;
    }
 
    // Driver code
    public static void Main()
    {
        int n = 5;
        Console.WriteLine(countNumbers(n));
    }
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

PHP

<?php
// PHP program to count numbers 
// with N digits which consists 
// of odd number of 0's
 
// Function to count Numbers 
// with N digits which consists
// of odd number of 0's
function countNumbers($N)
{
    return (pow(10, $N) - 
            pow(8, $N)) / 2;
}
 
// Driver code
$n = 5;
 
echo countNumbers($n) ;
 
// This code is contributed
// by Shivi_Aggarwal
?>

Javascript

<script>
    // Javascript program to count numbers with N digits
    // which consists of odd number of 0's
     
    // Function to count Numbers with N digits
    // which consists of odd number of 0's
    function countNumbers(N)
    {
        return parseInt((Math.pow(10, N) - Math.pow(8, N)) / 2, 10);
    }
     
    let n = 5;
    document.write(countNumbers(n));
     
    // This code is contributed by divyeshrabadiya07.
     
</script>

Output

Note: Answer can be very large, so for N greater than 9, use modular exponentiation.

Time Complexity: O(log n)

Auxiliary Space: O(1), since no extra space has been taken.

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