Find product of all elements at indexes which are factors of M for all possible sorted subsequences of length M

Given an array arr[] of N distinct integers and a positive integer M, the task is to find the product of all the elements at the indexes which are the factors of M for all the possible sorted subsequences of length M from the given array arr[].
Note: The product may be very large, take modulo to 10⁹ + 7.

Examples:

Input: arr[] = {4, 7, 5, 9, 3}, M = 4 
Output: 808556639 
Explanation:
There are five possible sets. They are: 
{4, 7, 5, 9}. In the sorted order, this set becomes {4, 5, 7, 9}. In this set, index 1, 2 and 4 divides M completely. Therefore, arr[1] * arr[2] * arr[4] = 4 * 5 * 9 = 180. 
Similarly, the remaining four sets along with their products are: 
{4, 7, 9, 3} -> 108 
{4, 5, 9, 3} -> 108 
{4, 7, 5, 3} -> 84 
{7, 5, 9, 3} -> 135 
The total value = ((180 * 108 * 108 * 84 * 135) % (10^9+7)) = 808556639

Input:arr[] = {7, 8, 9}, M = 2 
Output: 254016 

Approach: Since it is not possible to find all the sets, the idea is to count the total number of times every element occurs in the required position. If the count is found, then:  

product = (arr[1]^{count of arr[1]}) * (arr[2]^{count of arr[2]})* ….. *(arr[N]^{count of arr[N]}) 

• In order to find the count of the arr[i], we have to find the total number of the different sets that can be formed by placing arr[i] at every possible index(that divides M completely).
• Therefore, the number of sets formed by placing arr[i] at j^th(j divides m completely) index will be: 
  

^{(Number of elements lesser than arr[i])}C_j-1 * ^{(Number of elements greater than arr[i])} C_M-j

• As the count of any element may be very large, so for finding (arr[i]^{count of arr[i]}) % (10⁹ + 7), Fermat’s little theorem is used.

Below is the implementation of the above approach: 

C++

// C++ program to find the product of
// all the combinations of M elements
// from an array whose index in the
// sorted order divides M completely

#include <bits/stdc++.h>
using namespace std;

typedef long long int lli;
const int m = 4;

// Iterative Function to calculate
// (x^y)%p in O(log y)
long long int power(lli x, lli y, lli p)
{
    lli res = 1;
    x = x % p;

    while (y > 0) {

        // If y is odd, multiply x with result
        if (y & 1)
            res = (res * x) % p;

        // y must be even now
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}

// Iterative Function to calculate
// (nCr)%p and save in f[n][r]
// C(n, r)%p = [ C(n-1, r-1)%p + C(n-1, r)%p ] % p
// and C(n, 0) = C(n, n) = 1
void nCr(lli n, lli p, lli f[][m + 1])
{
    for (lli i = 0; i <= n; i++) {
        for (lli j = 0; j <= m; j++) {

            // If j>i then C(i, j) = 0
            if (j > i)
                f[i][j] = 0;

            // If i is equal to j then C(i, j) = 1
            else if (j == 0 || j == i)
                f[i][j] = 1;

            // C(i, j) = ( C(i-1, j) + C(i-1, j-1))%p
            else
                f[i][j] = (f[i - 1][j]
                           + f[i - 1][j - 1])
                          % p;
        }
    }
}

void operations(lli arr[], lli n, lli f[][m + 1])
{
    lli p = 1000000007;
    nCr(n, p - 1, f);

    sort(arr, arr + n);

    // Initialize the answer
    lli ans = 1;

    for (lli i = 0; i < n; i++) {

        // For every element arr[i],
        // x is count of occurrence
        // of arr[i] in different set
        // such that index of arr[i]
        // in those sets divides m completely.
        long long int x = 0;

        for (lli j = 1; j <= m; j++) {

            // Finding the count of arr[i]
            // by placing it at the index
            // which divides m completely
            if (m % j == 0)

                // Using fermat's little theorem
                x = (x
                     + (f[n - i - 1][m - j]
                        * f[i][j - 1])
                           % (p - 1))
                    % (p - 1);
        }

        // Multiplying with the count
        ans = ((ans * power(arr[i],
                            x, p))
               % p);
    }

    cout << ans << endl;
}

// Driver code
int main()
{

    lli arr[] = { 4, 5, 7, 9, 3 };

    lli n = sizeof(arr) / sizeof(arr[0]);

    lli f[n + 1][m + 1];

    operations(arr, n, f);
}

Java

// Java program to find the product of
// all the combinations of M elements
// from an array whose index in the
// sorted order divides M completely
import java.util.*;

class GFG{

static int m = 4;

// Iterative Function to calculate
// (x^y)%p in O(log y)
static long power(long x, long y, long p)
{
    long res = 1;
    x = x % p;

    while (y > 0)
    {

        // If y is odd, multiply
        // x with result
        if (y % 2 == 1)
            res = (res * x) % p;

        // y must be even now
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}

// Iterative Function to calculate
// (nCr)%p and save in f[n][r]
// C(n, r)%p = [ C(n-1, r-1)%p + C(n-1, r)%p ] % p
// and C(n, 0) = C(n, n) = 1
static void nCr(int n, long p, int f[][])
{
    for(int i = 0; i <= n; i++)
    {
       for(int j = 0; j <= m; j++)
       {

          // If j>i then C(i, j) = 0
          if (j > i)
              f[i][j] = 0;

          // If i is equal to j
          // then C(i, j) = 1
          else if (j == 0 || j == i)
              f[i][j] = 1;

          // C(i, j) = ( C(i-1, j) + C(i-1, j-1))%p
          else
              f[i][j] = (f[i - 1][j] +
                         f[i - 1][j - 1]) % (int)p;
       }
    }
}

static void operations(int arr[], int n, int f[][])
{
    long p = 1000000007;
    nCr(n, p - 1, f);

    Arrays.sort(arr);

    // Initialize the answer
    long ans = 1;

    for(int i = 0; i < n; i++)
    {

       // For every element arr[i],
       // x is count of occurrence
       // of arr[i] in different set
       // such that index of arr[i]
       // in those sets divides m
       // completely.
       long x = 0;

       for(int j = 1; j <= m; j++)
       {

          // Finding the count of arr[i]
          // by placing it at the index
          // which divides m completely
          if (m % j == 0)

              // Using fermat's little theorem
              x = (x + (f[n - i - 1][m - j] *
                               f[i][j - 1]) %
                                   (p - 1)) %
                                   (p - 1);
       }

       // Multiplying with the count
       ans = ((ans * power(arr[i], x, p)) % p);
    }
    System.out.print(ans + "\n");
}

// Driver code
public static void main(String[] args)
{
    int arr[] = { 4, 5, 7, 9, 3 };
    int n = arr.length;
    int [][]f = new int[n + 1][m + 1];

    operations(arr, n, f);
}
}

// This code is contributed by Rohit_ranjan

Python3

# Python3 program to find the product of
# all the combinations of M elements
# from an array whose index in the
# sorted order divides M completely
m = 4

# Iterative Function to calculate
# (x^y)%p in O(log y)
def power(x, y, p):

    res = 1
    x = x % p

    while (y > 0):

        # If y is odd, multiply x with result
        if (y & 1):
            res = (res * x) % p

        # y must be even now
        y = y >> 1
        x = (x * x) % p
    return res

# Iterative Function to calculate
# (nCr)%p and save in f[n][r]
# C(n, r)%p = [ C(n-1, r-1)%p +
# C(n-1, r)%p ] % p
# and C(n, 0) = C(n, n) = 1
def nCr(n, p, f):

    for i in range (n):
        for j in range (m + 1):

            # If j>i then C(i, j) = 0
            if (j > i):
                f[i][j] = 0

            # If i is equal to j then
            # C(i, j) = 1
            elif (j == 0 or j == i):
                f[i][j] = 1

            # C(i, j) = ( C(i-1, j) +
            # C(i-1, j-1))%p
            else:
                f[i][j] = ((f[i - 1][j] +
                            f[i - 1][j - 1]) % p)

def operations(arr, n, f):

    p = 1000000007
    nCr(n, p - 1, f)

    arr.sort()

    # Initialize the answer
    ans = 1

    for i in range (n):

        # For every element arr[i],
        # x is count of occurrence
        # of arr[i] in different set
        # such that index of arr[i]
        # in those sets divides m completely.
        x = 0

        for j in range (1, m + 1):

            # Finding the count of arr[i]
            # by placing it at the index
            # which divides m completely
            if (m % j == 0):

                # Using fermat's little theorem
                x = ((x + (f[n - i - 1][m - j] *
                           f[i][j - 1]) % (p - 1)) %
                                          (p - 1))

        # Multiplying with the count
        ans = ((ans * power(arr[i],
                            x, p)) % p)

    print (ans)

# Driver code
if __name__ == "__main__":
    arr = [4, 5, 7, 9, 3]
    n = len(arr)
    f = [[0 for x in range (m + 1)]
            for y in range (n + 1)]
    operations(arr, n, f)

# This code is contributed by Chitranayal

C#

// C# program to find the product of
// all the combinations of M elements
// from an array whose index in the
// sorted order divides M completely
using System;

class GFG{

static int m = 4;

// Iterative Function to calculate
// (x^y)%p in O(log y)
static long power(long x, long y, long p)
{
    long res = 1;
    x = x % p;

    while (y > 0)
    {

        // If y is odd, multiply
        // x with result
        if (y % 2 == 1)
            res = (res * x) % p;

        // y must be even now
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}

// Iterative Function to calculate
// (nCr)%p and save in f[n,r]
// C(n, r)%p = [ C(n-1, r-1)%p + C(n-1, r)%p ] % p
// and C(n, 0) = C(n, n) = 1
static void nCr(int n, long p, int [,]f)
{
    for(int i = 0; i <= n; i++)
    {
       for(int j = 0; j <= m; j++)
       {

          // If j>i then C(i, j) = 0
          if (j > i)
              f[i, j] = 0;

          // If i is equal to j
          // then C(i, j) = 1
          else if (j == 0 || j == i)
              f[i, j] = 1;

          // C(i, j) = ( C(i-1, j) + C(i-1, j-1))%p
          else
              f[i, j] = (f[i - 1, j] +
                         f[i - 1, j - 1]) % (int)p;
       }
    }
}

static void operations(int []arr, int n, int [,]f)
{
    long p = 1000000007;
    nCr(n, p - 1, f);

    Array.Sort(arr);

    // Initialize the answer
    long ans = 1;

    for(int i = 0; i < n; i++)
    {

       // For every element arr[i],
       // x is count of occurrence
       // of arr[i] in different set
       // such that index of arr[i]
       // in those sets divides m
       // completely.
       long x = 0;
       for(int j = 1; j <= m; j++)
       {

          // Finding the count of arr[i]
          // by placing it at the index
          // which divides m completely
          if (m % j == 0)

              // Using fermat's little theorem
              x = (x + (f[n - i - 1, m - j] *
                               f[i, j - 1]) %
                                   (p - 1)) %
                                   (p - 1);
       }

       // Multiplying with the count
       ans = ((ans * power(arr[i], x, p)) % p);
    }
    Console.Write(ans + "\n");
}

// Driver code
public static void Main(String[] args)
{
    int []arr = { 4, 5, 7, 9, 3 };
    int n = arr.Length;
    int [,]f = new int[n + 1, m + 1];

    operations(arr, n, f);
}
}

// This code is contributed by Rohit_ranjan

Javascript

<script>

// Javascript program to find the product of
// all the combinations of M elements
// from an array whose index in the
// sorted order divides M completely
let m = 4;

// Iterative Function to calculate
// (x^y)%p in O(log y)
function power(x, y, p)
{
    let res = 1;
    x = x % p;

    while (y > 0)
    {

        // If y is odd, multiply
        // x with result
        if (y % 2 == 1)
            res = (res * x) % p;

        // y must be even now
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}

// Iterative Function to calculate
// (nCr)%p and save in f[n][r]
// C(n, r)%p = [ C(n-1, r-1)%p + C(n-1, r)%p ] % p
// and C(n, 0) = C(n, n) = 1
function nCr(n, p, f)
{
    for(let i = 0; i <= n; i++)
    {
       for(let j = 0; j <= m; j++)
       {

          // If j>i then C(i, j) = 0
          if (j > i)
              f[i][j] = 0;

          // If i is equal to j
          // then C(i, j) = 1
          else if (j == 0 || j == i)
              f[i][j] = 1;

          // C(i, j) = ( C(i-1, j) + C(i-1, j-1))%p
          else
              f[i][j] = (f[i - 1][j] +
                         f[i - 1][j - 1]) % p;
       }
    }
}

function operations(arr, n, f)
{
    let p = 1000000007;
    nCr(n, p - 1, f);

    arr.sort();

    // Initialize the answer
    let ans = 1;

    for(let i = 0; i < n; i++)
    {

       // For every element arr[i],
       // x is count of occurrence
       // of arr[i] in different set
       // such that index of arr[i]
       // in those sets divides m
       // completely.
       let x = 0;

       for(let j = 1; j <= m; j++)
       {

          // Finding the count of arr[i]
          // by placing it at the index
          // which divides m completely
          if (m % j == 0)

              // Using fermat's little theorem
              x = (x + (f[n - i - 1][m - j] *
                                f[i][j - 1]) %
                                    (p - 1)) %
                                    (p - 1);
       }

       // Multiplying with the count
       ans = ((ans * power(arr[i], x, p)) % p);
    }
    document.write(ans + "\n");
}

// Driver Code
let arr = [ 4, 5, 7, 9, 3 ];
let n = arr.length;
let f = new Array(n + 1);

for(var i = 0; i < f.length; i++)
{
    f[i] = new Array(2);
}

operations(arr, n, f);

// This code is contributed by target_2

</script>

Output:

808556639

Time Complexity: O(N * M) where N is the length of the array and M is given by the user. 

Auxiliary Space: O(N * M)