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Probability of distributing M items among X bags such that first bag contains N items

  • Last Updated : 03 May, 2021

Given three integers N, M, X. The task is to find the probability of distributing M items among X bags such that first bag contains N items
Examples: 
 

Input : M = 7, X =3, N = 3 
Output : 0.2 
The Number of ways to keep 7 items in 3 bags is 6\choose 2
The Number of ways to keep 4 items in 2 bags is 3\choose 1  . As the first bag contains 3 items. 
The probability is 3\choose 1  /6\choose 2
Input : M = 9, X = 3, N = 4 
Output : 0.142857 
 

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Approach : 
In general, the Number of ways to place N items in K bags is N-1\choose K-1
 

  • The Number of ways to keep M items in X bags is M-1\choose X-1  .
  • The Number of ways to keep (M-N) items in (X-1) bags is M-N-1\choose X-2  . As the first bag contains N items.
  • The probability is M-N-1\choose X-2  /M-1\choose X-1  .

Below is the implementation of the above approach:
 

C++




// CPP program to find probability of
// first bag to contain N items such
// that M items are distributed among X bags
#include <bits/stdc++.h>
using namespace std;
 
// Function to find factorial of a number
int factorial(int n)
{
    if (n <= 1)
        return 1;
    return n * factorial(n - 1);
}
 
// Function to find nCr
int nCr(int n, int r)
{
    return factorial(n) / (factorial(r) * factorial(n - r));
}
 
// Function to find probability of
// first bag to contain N items such
// that M items are distributed among X bags
float Probability(int M, int N, int X)
{
    return (float)(nCr(M - N - 1, X - 2) /
                    (nCr(M - 1, X - 1) * 1.0));
}
 
// Driver code
int main()
{
    int M = 9, X = 3, N = 4;
 
    // Function call
    cout << Probability(M, N, X);
 
    return 0;
}

Java




// Java program to find probability of
// first bag to contain N items such
// that M items are distributed among X bags
 
class GFG
{
 
    // Function to find factorial of a number
    public static int factorial(int n)
    {
        if (n <= 1)
            return 1;
 
        return n * factorial(n - 1);
    }
 
    // Function to find nCr
    public static int nCr(int n, int r)
    {
        return factorial(n) / (factorial(r) * factorial(n - r));
    }
 
    // Function to find probability of
    // first bag to contain N items such
    // that M items are distributed among X bags
    public static float Probability(int M, int N, int X)
    {
        return (float) (nCr(M - N - 1, X - 2) / (nCr(M - 1, X - 1) * 1.0));
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int M = 9, X = 3, N = 4;
 
        // Function call
        System.out.println(Probability(M, N, X));
    }
}
 
// This code is contributed by
// sanjeev2552

Python3




# Python3 program to find probability of
# first bag to contain N items such
# that M items are distributed among X bags
 
# Function to find factorial of a number
def factorial(n) :
 
    if (n <= 1) :
        return 1;
         
    return n * factorial(n - 1);
 
# Function to find nCr
def nCr(n, r) :
 
    return (factorial(n) / (factorial(r) *
                            factorial(n - r)));
 
# Function to find probability of
# first bag to contain N items such
# that M items are distributed among X bags
def Probability(M, N, X) :
 
    return float(nCr(M - N - 1, X - 2) /
                (nCr(M - 1, X - 1) * 1.0));
 
# Driver code
if __name__ == "__main__" :
 
    M = 9; X = 3; N = 4;
 
    # Function call
    print(Probability(M, N, X));
 
# This code is contributed by AnkitRai01

C#




// C# program to find probability of
// first bag to contain N items such
// that M items are distributed among X bags
using System;
 
class GFG
{
  
    // Function to find factorial of a number
    static int factorial(int n)
    {
        if (n <= 1)
            return 1;
  
        return n * factorial(n - 1);
    }
  
    // Function to find nCr
    static int nCr(int n, int r)
    {
        return factorial(n) / (factorial(r) * factorial(n - r));
    }
  
    // Function to find probability of
    // first bag to contain N items such
    // that M items are distributed among X bags
    static float Probability(int M, int N, int X)
    {
        return (float) (nCr(M - N - 1, X - 2) / (nCr(M - 1, X - 1) * 1.0));
    }
  
    // Driver code
    static void Main()
    {
        int M = 9, X = 3, N = 4;
  
        // Function call
        Console.WriteLine(Probability(M, N, X));
    }
}
  
// This code is contributed by
// mohitkumar 29

Javascript




<script>
// Java Script program to find probability of
// first bag to contain N items such
// that M items are distributed among X bags
 
 
    // Function to find factorial of a number
    function factorial( n)
    {
        if (n <= 1)
            return 1;
 
        return n * factorial(n - 1);
    }
 
    // Function to find nCr
    function nCr( n,  r)
    {
        return factorial(n) / (factorial(r) * factorial(n - r));
    }
 
    // Function to find probability of
    // first bag to contain N items such
    // that M items are distributed among X bags
    function Probability(M,N,X)
    {
        return parseFloat(nCr(M - N - 1, X - 2) / (nCr(M - 1, X - 1) * 1.0));
    }
 
    // Driver code
    let M = 9, X = 3, N = 4;
 
        // Function call
        document.write(Probability(M, N, X).toFixed(6));
// This code is contributed by Bobby
</script>
Output: 
0.142857

 




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