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Number of pairs of arrays (A, B) such that A is ascending, B is descending and A[i] ≤ B[i]
• Difficulty Level : Hard
• Last Updated : 02 Jun, 2021

Given two integers N and M, the task is to find the number of pairs of arrays (A, B) such that array A and B both are of size M each where each entry of A and B is an integer between 1 and N such that for each i between 1 and M, A[i] ≤ B[i]. It is also given that the array A is sorted in non-descending order and B is sorted in non-ascending order. Since the answer can be very large, return answer modulo 109 + 7.

Examples:

Input: N = 2, M = 2
Output:
1: A= [1, 1] B=[1, 1]
2: A= [1, 1] B=[1, 2]
3: A= [1, 1] B=[2, 2]
4: A= [1, 2] B=[2, 2]
5: A= [2, 2] B=[2, 2]

Input: N = 5, M = 3
Output: 210

Approach: Notice that if there is a valid pair of arrays A and B and if B is concatenated after A the resultant array will always be either an ascending or a non-descending array of size of 2 * M. Each element of (A + B) will be between 1 and N (It is not necessary that all elements between 1 and N have to be used). This now simply converts the given problem to finding all the possible combinations of size 2 * M where each element is between 1 to N (with repetitions allowed) whose formula is 2 * M + N – 1CN – 1 or (2 * M + N – 1)! / ((2 * M)! * (N – 1)!).

Below is the implementation of the above approach:

## C++

 `// C++ code of above approach``#include ``#define mod 1000000007``using` `namespace` `std;` `long` `long` `fact(``long` `long` `n)``{``    ``if``(n == 1)``        ``return` `1;``    ``else``        ``return` `(fact(n - 1) * n) % mod;``}` `// Function to return the count of pairs``long` `long` `countPairs(``int` `m, ``int` `n)``{``    ``long` `long` `ans = fact(2 * m + n - 1) /``                    ``(fact(n - 1) * fact(2 * m));``    ``return` `(ans % mod);``}` `// Driver code``int` `main()``{``    ``int` `n = 5, m = 3;``    ``cout << (countPairs(m, n));``    ``return` `0;``}` `// This code is contributed by mohit kumar 29`

## Java

 `// Java code of above approach``class` `GFG``{``    ``final` `static` `long` `mod = ``1000000007` `;` `    ``static` `long` `fact(``long` `n)``    ``{``        ``if``(n == ``1``)``            ``return` `1``;``        ``else``            ``return` `(fact(n - ``1``) * n) % mod;``    ``}``    ` `    ``// Function to return the count of pairs``    ``static` `long` `countPairs(``int` `m, ``int` `n)``    ``{``        ``long` `ans = fact(``2` `* m + n - ``1``) /``                   ``(fact(n - ``1``) * fact(``2` `* m));``        ` `        ``return` `(ans % mod);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `main (String[] args)``    ``{``        ``int` `n = ``5``, m = ``3``;``        ` `        ``System.out.println(countPairs(m, n));``    ``}``}` `// This code is contributed by AnkitRai01`

## Python3

 `# Python3 implementation of the approach``from` `math ``import` `factorial as fact` `# Function to return the count of pairs``def` `countPairs(m, n):``    ``ans ``=` `fact(``2` `*` `m ``+` `n``-``1``)``/``/``(fact(n``-``1``)``*``fact(``2` `*` `m))``    ``return` `(ans ``%``(``10``*``*``9` `+` `7``))` `# Driver code``n, m ``=` `5``, ``3``print``(countPairs(m, n))`

## C#

 `// C# code of above approach``using` `System;` `class` `GFG``{``    ``static` `long` `mod = 1000000007 ;` `    ``static` `long` `fact(``long` `n)``    ``{``        ``if``(n == 1)``            ``return` `1;``        ``else``            ``return` `(fact(n - 1) * n) % mod;``    ``}``    ` `    ``// Function to return the count of pairs``    ``static` `long` `countPairs(``int` `m, ``int` `n)``    ``{``        ``long` `ans = fact(2 * m + n - 1) /``                ``(fact(n - 1) * fact(2 * m));``        ` `        ``return` `(ans % mod);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `n = 5, m = 3;``        ` `        ``Console.WriteLine(countPairs(m, n));``    ``}``}` `// This code is contributed by AnkitRai01`

## Javascript

 ``
Output:
`210`

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