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# Count number of subsets whose median is also present in the same subset

• Last Updated : 27 May, 2021

Given an array arr[] of size N, the task is to count the number of ways we can select a subset from the given array elements such that the median of the selected subset is also present as an element in the subset. Since this number may be large, compute it modulo 1000000007.
Examples:

Input: arr[] = {2, 3, 2}
Output:
{2}, {3}, {2}, {2, 2} and {2, 3, 2} are all possible valid sub-sets.
Input: arr[] = {1, 4, 2, 2, 3, 5}
Output: 36

Approach:

• Every subset of odd size has its median present in the subset, so, we can directly add 2N – 1 to the answer.
• For an even size subset, The subset will be selected, if and only if the middle two elements are equal.
• We need to count the number of even-sized subsets with equal middle elements.

The Simple solution would be to iterate over every pair of equal elements (i, j) such that A[i] = A[j] and iterate over the size 2 * X of a subset from X = 1 to N. The number of ways to make the subset of size X with two fixed middle elements is just the product of the number of ways we can select X – 1 element from [1, i – 1] and X – 1 element from [j + 1, N].
This solution requires to iterate over every pair (i, j) which takes O(N2) time and O(N) time per pair, leading to Overall time complexity O(N3
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#include ``using` `namespace` `std;``long` `long` `mod = 1000000007;` `// Function to return the factorial of a number``int` `fact(``int` `n)``{``    ``int` `res = 1;``    ``for` `(``int` `i = 2; i <= n; i++)``        ``res = res * i;``    ``return` `res;``}` `// Function to return the value of nCr``int` `nCr(``int` `n, ``int` `r)``{``    ``return` `fact(n) / (fact(r) * fact(n - r));``}` `// Function to return a raised to the power n``// with complexity O(log(n))``long` `long` `powmod(``long` `long` `a, ``long` `long` `n)``{``    ``if` `(!n)``        ``return` `1;``    ``long` `long` `pt = powmod(a, n / 2);``    ``pt = (pt * pt) % mod;``    ``if` `(n % 2)``        ``return` `(pt * a) % mod;``    ``else``        ``return` `pt;``}` `// Function to return the number of sub-sets``// whose median is also present in the set``long` `long` `CountSubset(``int``* arr, ``int` `n)``{` `    ``// Number of odd length sub-sets``    ``long` `long` `ans = powmod(2, n - 1);` `    ``// Sort the array``    ``sort(arr, arr + n);``    ``for` `(``int` `i = 0; i < n; ++i) {``        ``int` `j = i + 1;` `        ``// Checking each element for leftmost middle``        ``// element while they are equal``        ``while` `(j < n && arr[j] == arr[i]) {` `            ``// Calculate the number of elements in``            ``// right of rightmost middle element``            ``int` `r = n - 1 - j;` `            ``// Calculate the number of elements in``            ``// left of leftmost middle element``            ``int` `l = i;` `            ``// Add selected even length subsets``            ``// to the answer``            ``ans = (ans + nCr(l + r, l)) % mod;``            ``j++;``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``int` `arr[] = { 2, 3, 2 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);``    ``cout << CountSubset(arr, n) << endl;` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG {` `    ``static` `long` `mod = ``1000000007``;` `    ``// Function to return the factorial of a number``    ``static` `int` `fact(``int` `n)``    ``{``        ``int` `res = ``1``;``        ``for` `(``int` `i = ``2``; i <= n; i++)``            ``res = res * i;``        ``return` `res;``    ``}` `    ``// Function to return the value of nCr``    ``static` `int` `nCr(``int` `n, ``int` `r)``    ``{``        ``return` `fact(n) / (fact(r) * fact(n - r));``    ``}` `    ``// Function to return a raised to the power n``    ``// with complexity O(log(n))``    ``static` `long` `powmod(``long` `a, ``long` `n)``    ``{``        ``if` `(n == ``0``)``            ``return` `1``;``        ``long` `pt = powmod(a, n / ``2``);``        ``pt = (pt * pt) % mod;``        ``if` `(n % ``2` `== ``1``)``            ``return` `(pt * a) % mod;``        ``else``            ``return` `pt;``    ``}` `    ``// Function to return the number of sub-sets``    ``// whose median is also present in the set``    ``static` `long` `CountSubset(``int``[] arr, ``int` `n)``    ``{` `        ``// Number of odd length sub-sets``        ``long` `ans = powmod(``2``, n - ``1``);` `        ``// Sort the array``        ``Arrays.sort(arr);``        ``for` `(``int` `i = ``0``; i < n; ++i) {``            ``int` `j = i + ``1``;` `            ``// Checking each element for leftmost middle``            ``// element while they are equal``            ``while` `(j < n && arr[j] == arr[i]) {` `                ``// Calculate the number of elements in``                ``// right of rightmost middle element``                ``int` `r = n - ``1` `- j;` `                ``// Calculate the number of elements in``                ``// left of leftmost middle element``                ``int` `l = i;` `                ``// Add selected even length subsets``                ``// to the answer``                ``ans = (ans + nCr(l + r, l)) % mod;``                ``j++;``            ``}``        ``}` `        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `arr[] = { ``2``, ``3``, ``2` `};``        ``int` `n = arr.length;``        ``System.out.println(CountSubset(arr, n));``    ``}``}` `// This code has been contributed by 29AjayKumar`

## Python3

 `# Python 3 implementation of the approach``mod ``=` `1000000007` `# Function to return``# the factorial of a number``def` `fact(n):``    ``res ``=` `1``    ``for` `i ``in` `range``(``2``, n ``+` `1``):``        ``res ``=` `res ``*` `i``    ``return` `res` `# Function to return the value of nCr``def` `nCr(n, r):``    ``return` `int``(fact(n) ``/` `(fact(r) ``*``                          ``fact(n ``-` `r)))` `# Function to return 'a' raised to the power n``# with complexity O(log(n))``def` `powmod(a, n):``    ``if` `(n ``=``=` `0``):``        ``return` `1``    ``pt ``=` `powmod(a, ``int``(n ``/` `2``))``    ``pt ``=` `(pt ``*` `pt) ``%` `mod``    ``if` `(n ``%` `2``):``        ``return` `(pt ``*` `a) ``%` `mod``    ``else``:``        ``return` `pt` `# Function to return the number of sub-sets``# whose median is also present in the set``def` `CountSubset(arr, n):``    ` `    ``# Number of odd length sub-sets``    ``ans ``=` `powmod(``2``, n ``-` `1``)` `    ``# Sort the array``    ``arr.sort(reverse ``=` `False``)``    ``for` `i ``in` `range``(n):``        ``j ``=` `i ``+` `1` `        ``# Checking each element for leftmost middle``        ``# element while they are equal``        ``while` `(j < n ``and` `arr[j] ``=``=` `arr[i]):``            ` `            ``# Calculate the number of elements in``            ``# right of rightmost middle element``            ``r ``=` `n ``-` `1` `-` `j` `            ``# Calculate the number of elements in``            ``# left of leftmost middle element``            ``l ``=` `i` `            ``# Add selected even length subsets``            ``# to the answer``            ``ans ``=` `(ans ``+` `nCr(l ``+` `r, l)) ``%` `mod``            ``j ``+``=` `1` `    ``return` `ans` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``arr ``=` `[``2``, ``3``, ``2``]``    ``n ``=` `len``(arr)``    ``print``(CountSubset(arr, n))` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG {` `    ``static` `long` `mod = 1000000007;` `    ``// Function to return the factorial of a number``    ``static` `int` `fact(``int` `n)``    ``{``        ``int` `res = 1;``        ``for` `(``int` `i = 2; i <= n; i++)``            ``res = res * i;``        ``return` `res;``    ``}` `    ``// Function to return the value of nCr``    ``static` `int` `nCr(``int` `n, ``int` `r)``    ``{``        ``return` `fact(n) / (fact(r) * fact(n - r));``    ``}` `    ``// Function to return a raised to the power n``    ``// with complexity O(log(n))``    ``static` `long` `powmod(``long` `a, ``long` `n)``    ``{``        ``if` `(n == 0)``            ``return` `1;``        ``long` `pt = powmod(a, n / 2);``        ``pt = (pt * pt) % mod;``        ``if` `(n % 2 == 1)``            ``return` `(pt * a) % mod;``        ``else``            ``return` `pt;``    ``}` `    ``// Function to return the number of sub-sets``    ``// whose median is also present in the set``    ``static` `long` `CountSubset(``int``[] arr, ``int` `n)``    ``{` `        ``// Number of odd length sub-sets``        ``long` `ans = powmod(2, n - 1);` `        ``// Sort the array``        ``Array.Sort(arr);``        ``for` `(``int` `i = 0; i < n; ++i) {``            ``int` `j = i + 1;` `            ``// Checking each element for leftmost middle``            ``// element while they are equal``            ``while` `(j < n && arr[j] == arr[i]) {` `                ``// Calculate the number of elements in``                ``// right of rightmost middle element``                ``int` `r = n - 1 - j;` `                ``// Calculate the number of elements in``                ``// left of leftmost middle element``                ``int` `l = i;` `                ``// Add selected even length subsets``                ``// to the answer``                ``ans = (ans + nCr(l + r, l)) % mod;``                ``j++;``            ``}``        ``}` `        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``int``[] arr = { 2, 3, 2 };``        ``int` `n = arr.Length;``        ``Console.WriteLine(CountSubset(arr, n));``    ``}``}` `// This code has been contributed by 29AjayKumar`

## PHP

 `

## Javascript

 ``
Output:
`5`

Time Complexity: O(N3)
The time complexity of the above approach can be reduced to O(N2) if we store pascal triangle in 2-d array. So, now we don’t have to calculate factorial again and again. Read more about Pascal triangle here.
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#include ``using` `namespace` `std;``long` `long` `mod = 1000000007;``long` `long` `arr;` `// Function to store pascal triangle in 2-d array``void` `Preprocess()``{``    ``arr = 1;``    ``for` `(``int` `i = 1; i <= 1000; ++i) {``        ``arr[i] = 1;``        ``for` `(``int` `j = 1; j < i; ++j) {``            ``arr[i][j] = (arr[i - 1][j - 1] + arr[i - 1][j]) % mod;``        ``}``        ``arr[i][i] = 1;``    ``}``}` `// Function to return a raised to the power n``// with complexity O(log(n))``long` `long` `powmod(``long` `long` `a, ``long` `long` `n)``{``    ``if` `(!n)``        ``return` `1;``    ``long` `long` `pt = powmod(a, n / 2);``    ``pt = (pt * pt) % mod;``    ``if` `(n % 2)``        ``return` `(pt * a) % mod;``    ``else``        ``return` `pt;``}` `// Function to return the number of sub-sets``// whose median is also present in the set``long` `long` `CountSubset(``int``* val, ``int` `n)``{` `    ``// Number of odd length sub-sets``    ``long` `long` `ans = powmod(2, n - 1);` `    ``// Sort the array``    ``sort(val, val + n);``    ``for` `(``int` `i = 0; i < n; ++i) {``        ``int` `j = i + 1;` `        ``// Checking each element for leftmost middle``        ``// element while they are equal``        ``while` `(j < n && val[j] == val[i]) {` `            ``// Calculate the number of elements in``            ``// right of rightmost middle element``            ``int` `r = n - 1 - j;` `            ``// Calculate the number of elements in``            ``// left of leftmost middle element``            ``int` `l = i;` `            ``// Add selected even length subsets``            ``// to the answer``            ``ans = (ans + arr[l + r][l]) % mod;``            ``j++;``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``Preprocess();``    ``int` `val[] = { 2, 3, 2 };``    ``int` `n = ``sizeof``(val) / ``sizeof``(val);``    ``cout << CountSubset(val, n) << endl;` `    ``return` `0;``}`

## Java

 `// Java implementation of the above approach``import` `java.util.Arrays;` `class` `GFG``{` `    ``static` `long` `mod = ``1000000007``;``    ``static` `long``[][] arr = ``new` `long``[``1001``][``1001``];` `    ``// Function to store pascal triangle in 2-d array``    ``static` `void` `Preprocess()``    ``{``        ``arr[``0``][``0``] = ``1``;``        ``for` `(``int` `i = ``1``; i <= ``1000``; ++i)``        ``{``            ``arr[i][``0``] = ``1``;``            ``for` `(``int` `j = ``1``; j < i; ++j)``            ``{``                ``arr[i][j] = (arr[i - ``1``][j - ``1``] + arr[i - ``1``][j]) % mod;``            ``}``            ``arr[i][i] = ``1``;``        ``}``    ``}` `    ``// Function to return a raised to the power n``    ``// with complexity O(log(n))``    ``static` `long` `powmod(``long` `a, ``long` `n)``    ``{``        ``if` `(n == ``0``)``        ``{``            ``return` `1``;``        ``}``        ``long` `pt = powmod(a, n / ``2``);``        ``pt = (pt * pt) % mod;``        ``if` `(n % ``2` `== ``1``)``        ``{``            ``return` `(pt * a) % mod;``        ``}``        ``else``        ``{``            ``return` `pt;``        ``}``    ``}` `    ``// Function to return the number of sub-sets``    ``// whose median is also present in the set``    ``static` `long` `CountSubset(``int``[] val, ``int` `n)``    ``{` `        ``// Number of odd length sub-sets``        ``long` `ans = powmod(``2``, n - ``1``);` `        ``// Sort the array``        ``Arrays.sort(val);``        ``for` `(``int` `i = ``0``; i < n; ++i)``        ``{``            ``int` `j = i + ``1``;` `            ``// Checking each element for leftmost middle``            ``// element while they are equal``            ``while` `(j < n && val[j] == val[i])``            ``{` `                ``// Calculate the number of elements in``                ``// right of rightmost middle element``                ``int` `r = n - ``1` `- j;` `                ``// Calculate the number of elements in``                ``// left of leftmost middle element``                ``int` `l = i;` `                ``// Add selected even length subsets``                ``// to the answer``                ``ans = (ans + arr[l + r][l]) % mod;``                ``j++;``            ``}``        ``}` `        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``Preprocess();``        ``int` `val[] = {``2``, ``3``, ``2``};``        ``int` `n = val.length;` `        ``System.out.println(CountSubset(val, n));``    ``}``}` `// This code contributed by Rajput-Ji`

## Python3

 `# Python3 implementation of the approach``mod ``=` `1000000007``arr ``=` `[[``None` `for` `i ``in` `range``(``1001``)] ``for` `j ``in` `range``(``1001``)]` `# Function to store pascal triangle in 2-d array``def` `Preprocess():`` ` `    ``arr[``0``][``0``] ``=` `1``    ``for` `i ``in` `range``(``1``, ``1001``): ``        ``arr[i][``0``] ``=` `1``        ``for` `j ``in` `range``(``1``, i): ``            ``arr[i][j] ``=` `(arr[i ``-` `1``][j ``-` `1``] ``+` `arr[i ``-` `1``][j]) ``%` `mod``         ` `        ``arr[i][i] ``=` `1``     ` `# Function to return a raised to the power n``# with complexity O(log(n))``def` `powmod(a, n):`` ` `    ``if` `not` `n:``        ``return` `1``    ``pt ``=` `powmod(a, n ``/``/` `2``)``    ``pt ``=` `(pt ``*` `pt) ``%` `mod``    ``if` `n ``%` `2``:``        ``return` `(pt ``*` `a) ``%` `mod``    ``else``:``        ``return` `pt` `# Function to return the number of sub-sets``# whose median is also present in the set``def` `CountSubset(val, n):`` ` `    ``# Number of odd length sub-sets``    ``ans ``=` `powmod(``2``, n ``-` `1``)` `    ``# Sort the array``    ``val.sort()``    ``for` `i ``in` `range``(``0``, n): ``        ``j ``=` `i ``+` `1` `        ``# Checking each element for leftmost middle``        ``# element while they are equal``        ``while` `j < n ``and` `val[j] ``=``=` `val[i]: ` `            ``# Calculate the number of elements in``            ``# right of rightmost middle element``            ``r ``=` `n ``-` `1` `-` `j` `            ``# Calculate the number of elements in``            ``# left of leftmost middle element``            ``l ``=` `i` `            ``# Add selected even length``            ``# subsets to the answer``            ``ans ``=` `(ans ``+` `arr[l ``+` `r][l]) ``%` `mod``            ``j ``+``=` `1` `    ``return` `ans`` ` `# Driver code``if` `__name__ ``=``=` `"__main__"``:`` ` `    ``Preprocess()``    ``val ``=` `[``2``, ``3``, ``2``]``    ``n ``=` `len``(val)``    ``print``(CountSubset(val, n))` `# This code is contributed by Rituraj Jain`

## C#

 `// C# implementation of the above approach``using` `System;``    ` `class` `GFG``{` `    ``static` `long` `mod = 1000000007;``    ``static` `long` `[,]arr = ``new` `long``[1001,1001];` `    ``// Function to store pascal triangle in 2-d array``    ``static` `void` `Preprocess()``    ``{``        ``arr[0,0] = 1;``        ``for` `(``int` `i = 1; i <= 1000; ++i)``        ``{``            ``arr[i,0] = 1;``            ``for` `(``int` `j = 1; j < i; ++j)``            ``{``                ``arr[i,j] = (arr[i - 1,j - 1] + arr[i - 1,j]) % mod;``            ``}``            ``arr[i,i] = 1;``        ``}``    ``}` `    ``// Function to return a raised to the power n``    ``// with complexity O(log(n))``    ``static` `long` `powmod(``long` `a, ``long` `n)``    ``{``        ``if` `(n == 0)``        ``{``            ``return` `1;``        ``}``        ``long` `pt = powmod(a, n / 2);``        ``pt = (pt * pt) % mod;``        ``if` `(n % 2 == 1)``        ``{``            ``return` `(pt * a) % mod;``        ``}``        ``else``        ``{``            ``return` `pt;``        ``}``    ``}` `    ``// Function to return the number of sub-sets``    ``// whose median is also present in the set``    ``static` `long` `CountSubset(``int``[] val, ``int` `n)``    ``{` `        ``// Number of odd length sub-sets``        ``long` `ans = powmod(2, n - 1);` `        ``// Sort the array``        ``Array.Sort(val);``        ``for` `(``int` `i = 0; i < n; ++i)``        ``{``            ``int` `j = i + 1;` `            ``// Checking each element for leftmost middle``            ``// element while they are equal``            ``while` `(j < n && val[j] == val[i])``            ``{` `                ``// Calculate the number of elements in``                ``// right of rightmost middle element``                ``int` `r = n - 1 - j;` `                ``// Calculate the number of elements in``                ``// left of leftmost middle element``                ``int` `l = i;` `                ``// Add selected even length subsets``                ``// to the answer``                ``ans = (ans + arr[l + r,l]) % mod;``                ``j++;``            ``}``        ``}` `        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``Preprocess();``        ``int` `[]val = {2, 3, 2};``        ``int` `n = val.Length;` `        ``Console.WriteLine(CountSubset(val, n));``    ``}``}` `/* This code contributed by PrinciRaj1992 */`

## Javascript

 ``
Output:
`5`

Time Complexity: O(N2)

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