# 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++

 `// CPP Program to find number of ways ` `// to merge two array such that their  ` `// order in merged array is same ` `#include ` `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; ` `} `

## 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. `

## Python3

 `# Python 3 Program to find number of ways ` `# to merge two array such that their  ` `# order in merged array is same ` ` `  `# function to find the binomial coefficient ` `def` `binomialCoeff(n, k): ` `    ``C ``=` `[``0` `for` `i ``in` `range``(k ``+` `1``)] ` ` `  `    ``C[``0``] ``=` `1` ` `  `    ``for` `i ``in` `range``(``1``, n ``+` `1``, ``1``): ` `         `  `        ``# Compute next row of pascal  ` `        ``# triangle using the previous row ` `        ``j ``=` `min``(i, k) ` `        ``while``(j > ``0``): ` `            ``C[j] ``=` `C[j] ``+` `C[j ``-` `1``] ` `            ``j ``-``=` `1` ` `  `    ``return` `C[k] ` ` `  `# function to find number of ways  ` `# to merge two array such that their  ` `# order in merged array is same ` `def` `numOfWays(n, m): ` `    ``return` `binomialCoeff(m ``+` `n, m) ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``n ``=` `2` `    ``m ``=` `2` `    ``print``(numOfWays(n, m)) ` `     `  `# This code is contributed by ` `# Sahil_shelangia `

## C#

 `// 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. `

## PHP

 ` 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  ` `function` `numOfWays( ``\$n``, ``\$m``)  ` `{  ` `    ``return` `binomialCoeff(``\$m` `+ ``\$n``, ``\$m``);  ` `}  ` ` `  `    ``\$n` `= 2; ``\$m` `= 2;  ` `    ``echo` `numOfWays(``\$n``, ``\$m``);  ` `   ``//This code is contributed by Rajput-Ji. ` `?> `

Output:

```6
```

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

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Improved By : vt_m, sahilshelangia, Rajput-Ji