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Total number of ways to place X and Y at n places such that no two X are together

  • Difficulty Level : Easy
  • Last Updated : 31 Mar, 2021

Given N positions, the task is to count the total number of ways to place X and Y such that no two X are together. 
Examples: 
 

Input: 3
Output: 5
XYX, YYX, YXY, XYY and YYY 

Input: 4
Output: 8
XYXY, XYYX, YXYX, YYYX, YYXY, YXYY, XYYY and YYYY

 

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Approach: 
For N = 1, X and Y are 2 possible ways. 
For N = 2, XY, YX and YY are the 3 possible ways. 
For N = 3, XYX, YYX, YXY, XYY and YYY are 5 possible ways. 
For N = 4, XYXY, XYYX, YXYX, YYYX, YYXY, YXYY, XYYY and YYYY are 8 possible ways. 
On solving for values of N, a Fibonacci pattern series is observed.
Below is the iterative implementation of the above approach: 
 

C++




// C++ program to find the
// number of ways Calculate
// total ways to place 'x'
// and 'y' at n places such
// that no two 'x' are together
#include <iostream>
using namespace std;
 
    // Function to return
    // number of ways
    int ways(int n)
    {
        // for n=1
        int first = 2;
 
        // for n=2
        int second = 3;
        int res = 0;
 
        // iterate to find
        // Fibonacci term
        for (int i = 3; i <= n; i++)
        {
            res = first + second;
            first = second;
            second = res;
        }
 
        return res;
    }
 
// Driver Code
int main()
{
    // total number of places
    int n = 7;
    cout << "Total ways are : ";
    cout << ways(n);
 
    return 0;
}
 
// This code is contributed
// by jit_t

Java




// Java program to find the number of ways
// Calculate total ways to place 'x' and 'y'
// at n places such that no two 'x' are together
public class GFG {
     
    // Function to return number of ways
    static int ways(int n)
    {
        // for n=1
        int first = 2;
 
        // for n=2
        int second = 3;
        int res = 0;
 
        // iterate to find Fibonacci term
        for (int i = 3; i <= n; i++) {
            res = first + second;
            first = second;
            second = res;
        }
 
        return res;
    }
    public static void main(String[] args)
    {
 
        // total number of places
        int n = 7;
 
        System.out.print("Total ways are: " + ways(n));
    }
}

Python3




# Python3 program to find the
# number of ways Calculate
# total ways to place 'x'
# and 'y' at n places such
# that no two 'x' are together
 
# Function to return
# number of ways
def ways(n):
     
    # for n=1
    first = 2;
 
    # for n=2
    second = 3;
    res = 0;
 
    # iterate to find
    # Fibonacci term
    for i in range(3, n + 1):
        res = first + second;
        first = second;
        second = res;
 
    return res;
     
# Driver Code
 
# total number of places
n = 7;
print("Total ways are: " ,
                 ways(n));
 
# This code is contributed by mits

C#




// C# program to find the
// number of ways. Calculate
// total ways to place 'x'
// and 'y' at n places such
// that no two 'x' are together
using System;
 
class GFG
{
     
    // Function to return
    // number of ways
    static int ways(int n)
    {
        // for n=1
        int first = 2;
 
        // for n=2
        int second = 3;
        int res = 0;
 
        // iterate to find
        // Fibonacci term
        for (int i = 3; i <= n; i++)
        {
            res = first + second;
            first = second;
            second = res;
        }
 
        return res;
    }
     
    // Driver Code
    static public void Main ()
    {
         
        // total number
        // of places
        int n = 7;
 
        Console.WriteLine("Total ways are: " +
                                     ways(n));
    }
}
 
//This code is contributed by ajit

PHP




<?php
// PHP program to find the
// number of ways Calculate
// total ways to place 'x'
// and 'y' at n places such
// that no two 'x' are together
 
// Function to return
// number of ways
function ways($n)
{
    // for n=1
    $first = 2;
 
    // for n=2
    $second = 3;
    $res = 0;
 
    // iterate to find
    // Fibonacci term
    for ($i = 3; $i <= $n; $i++)
    {
        $res = $first + $second;
        $first = $second;
        $second = $res;
    }
 
    return $res;
}
// Driver Code
 
// total number of places
$n = 7;
echo "Total ways are: " , ways($n);
 
// This code is contributed by ajit
?>

Javascript




<script>
// javascript program to find the number of ways
// Calculate total ways to place 'x' and 'y'
// at n places such that no two 'x' are together
 
    // Function to return number of ways
    function ways(n)
    {
     
        // for n = 1
        var first = 2;
 
        // for n = 2
        var second = 3;
        var res = 0;
 
        // iterate to find Fibonacci term
        for (i = 3; i <= n; i++)
        {
            res = first + second;
            first = second;
            second = res;
        }
        return res;
    }
 
        // total number of places
        var n = 7;
        document.write("Total ways are: " + ways(n));
 
// This code is contributed by aashish1995
</script>
Output: 
Total ways are: 34

 

Time Complexity: O(N)
 




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