Number of ways to merge two arrays such retaining order

Given two array of size n and m. The task is to find the number of ways we can merge the given arrays into one array such that order of elements of each array doesn’t change.

Examples:

Input : n = 2, m = 2
Output : 6
Let first array of size n = 2 be [1, 2] and second array of size m = 2 be [3, 4].
So, possible merged array of n + m elements can be:
[1, 2, 3, 4]
[1, 3, 2, 4]
[3, 4, 1, 2]
[3, 1, 4, 2]
[1, 3, 4, 2]
[3, 1, 2, 4]
 
Input : n = 4, m = 6
Output : 210

The idea is to use the concept of combinatorics. Suppose we have two array A{a1, a2, …., am} and B{b1, b2, …., bn} having m and n elements respectively and now we have to merge them without loosing their order.
After merging we know that the total number of element will be (m + n) element after merging. So, now we just need the ways to choose m places out of (m + n) where you will place element of array A in its actual order, which is m + nCn.
After placing m element of array A, n spaces will be left, which can be filled by the n elements of B array in its actual order.
So, total number of ways to merge two array such that their order in merged array is same is m + nCn

Below is the implementation of this appraoch:

C++


[sourcecode language=”CPP”]
// CPP Program to find number of ways
// to merge two array such that their
// order in merged array is same
#include <bits/stdc++.h>
using namespace std;

// function to find the binomial coefficient
int binomialCoeff(int n, int k)
{
int C[k + 1];
memset(C, 0, sizeof(C));

C[0] = 1; // nC0 is 1

for (int i = 1; i <= n; i++) {

// Compute next row of pascal triangle
// using the previous row
for (int j = min(i, k); j > 0; j–)
C[j] = C[j] + C[j – 1];
}
return C[k];
}

// function to find number of ways
// to merge two array such that their
// order in merged array is same
int numOfWays(int n, int m)
{
return binomialCoeff(m + n, m);
}

// Driven Program
int main()
{
int n = 2, m = 2;
cout << numOfWays(n, m) << endl;
return 0;
}
[/sourcecode]

Java


[sourcecode language=”Java”]
// Java Program to find number of ways
// to merge two array such that their
// order in merged array is same

import java.io.*;

class GFG {

// function to find the binomial
// coefficient
static int binomialCoeff(int n, int k)
{
int C[] = new int[k + 1];
// memset(C, 0, sizeof(C));

C[0] = 1; // nC0 is 1

for (int i = 1; i <= n; i++) {

// Compute next row of pascal
// triangle using the previous
// row
for (int j = Math.min(i, k);
j > 0; j–)
C[j] = C[j] + C[j – 1];
}

return C[k];
}

// function to find number of ways
// to merge two array such that their
// order in merged array is same
static int numOfWays(int n, int m)
{
return binomialCoeff(m + n, m);
}

// Driven Program
public static void main (String[] args)
{
int n = 2, m = 2;
System.out.println(numOfWays(n, m));
}
}

// This code is contributed by anuj_67.
[/sourcecode]

C#


[sourcecode language=”CSHARP”]
// C# Program to find number of ways
// to merge two array such that their
// order in merged array is same

using System;

class GFG {

// function to find the binomial
// coefficient
static int binomialCoeff(int n, int k)
{
int []C = new int[k + 1];
// memset(C, 0, sizeof(C));

C[0] = 1; // nC0 is 1

for (int i = 1; i <= n; i++) {

// Compute next row of pascal
// triangle using the previous
// row
for (int j = Math.Min(i, k);
j > 0; j–)
C[j] = C[j] + C[j – 1];
}

return C[k];
}

// function to find number of ways
// to merge two array such that their
// order in merged array is same
static int numOfWays(int n, int m)
{
return binomialCoeff(m + n, m);
}

// Driven Program
public static void Main ()
{
int n = 2, m = 2;
Console.WriteLine(numOfWays(n, m));
}
}

// This code is contributed by anuj_67.

[/sourcecode]

Output:

6

We can solve above problem in linear time using linear time implementation of binomial coefficient.



My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Improved By : vt_m