Find a pair from the given array with maximum nCr value

Difficulty Level : Medium
Last Updated : 08 Jun, 2021

Given an array arr[] of n positive integers. The task is to find elements arr[i] and arr[j] from the array such that ^arr[i]C_arr[j] is maximum possible. In case of more than 1 valid pairs, print any one of them.
Examples:

Input: arr[] = {3, 1, 2}
Output: 3 2
³C₁ = 3
³C₂ = 3
²C₁ = 2
(3, 1) and (3, 2) are the only pairs with maximum ⁿC_r.
Input: arr[] = {9, 2, 4, 11, 6}
Output: 11 6

Approach: ⁿC_r is a monotonic increasing function, that is ^{n + 1}C_r > ⁿC_r. We can use this fact to get close to our answer, we will choose the max n among all the given integers. This way we fixed the value of n.
Now, we have to look for r. Since we know that ⁿC_r = ⁿC_{n – r}, it means ⁿC_r will first reach its maxima and then decrease.
If n is be odd, then our maxima will occur at n / 2 and n / 2 + 1.
For n = 11, we will get the maxima at ¹¹C₅ and ¹¹C₆.
When n is even, then our maxima will occur at n / 2.
Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to print the pair that gives maximum nCr
void printMaxValPair(vector<long long>& v, int n)
{
    sort(v.begin(), v.end());
 
    // This gives the value of N in nCr
    long long N = v[n - 1];
 
    // Case 1 : When N is odd
    if (N % 2 == 1) {
        long long first_maxima = N / 2;
        long long second_maxima = first_maxima + 1;
        long long ans1 = 3e18, ans2 = 3e18;
        long long from_left = -1, from_right = -1;
        long long from = -1;
        for (long long i = 0; i < n; ++i) {
            if (v[i] > first_maxima) {
                from = i;
                break;
            }
            else {
                long long diff = first_maxima - v[i];
                if (diff < ans1) {
                    ans1 = diff;
                    from_left = v[i];
                }
            }
        }
        from_right = v[from];
        long long diff1 = first_maxima - from_left;
        long long diff2 = from_right - second_maxima;
 
        if (diff1 < diff2)
            cout << N << " " << from_left;
        else
            cout << N << " " << from_right;
    }
 
    // Case 2 : When N is even
    else {
        long long maxima = N / 2;
        long long ans1 = 3e18;
        long long R = -1;
        for (long long i = 0; i < n - 1; ++i) {
            long long diff = abs(v[i] - maxima);
            if (diff < ans1) {
                ans1 = diff;
                R = v[i];
            }
        }
        cout << N << " " << R;
    }
}
 
// Driver code
int main()
{
    vector<long long> v = { 1, 1, 2, 3, 6, 1 };
    int n = v.size();
    printMaxValPair(v, n);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
 
class GFG
{
 
// Function to print the pair that gives maximum nCr
static void printMaxValPair(Vector<Long> v, int n)
{
    Collections.sort(v);
 
    // This gives the value of N in nCr
    long N = v.get((int)n - 1);
 
    // Case 1 : When N is odd
    if (N % 2 == 1)
    {
        long first_maxima = N / 2;
        long second_maxima = first_maxima + 1;
        long ans1 =(long) 3e18, ans2 = (long)3e18;
        long from_left = -1, from_right = -1;
        long from = -1;
        for (long i = 0; i < n; ++i)
        {
            if (v.get((int)i) > first_maxima)
            {
                from = i;
                break;
            }
            else
            {
                long diff = first_maxima - v.get((int)i);
                if (diff < ans1)
                {
                    ans1 = diff;
                    from_left = v.get((int)i);
                }
            }
        }
        from_right = v.get((int)from);
        long diff1 = first_maxima - from_left;
        long diff2 = from_right - second_maxima;
 
        if (diff1 < diff2)
            System.out.println( N + " " + from_left);
        else
            System.out.println( N + " " + from_right);
    }
 
    // Case 2 : When N is even
    else
    {
        long maxima = N / 2;
        long ans1 =(int) 3e18;
        long R = -1;
        for (long i = 0; i < n - 1; ++i)
        {
            long diff = Math.abs(v.get((int)i) - maxima);
             
            if (diff < ans1)
            {
                ans1 = diff;
                R = v.get((int)i);
            }
        }
        System.out.println( N + " " + R);
    }
}
 
// Driver code
public static void main(String args[])
{
    long arr[] = { 1, 1, 2, 3, 6, 1};
    Vector<Long> v = new Vector<Long>( );
     
    for(int i = 0; i < arr.length; i++)
    v.add(arr[i]);
     
    int n = v.size();
    printMaxValPair(v, n);
}
}
 
// This code is contributed by Arnab Kundu

Python3

# Python3 implementation of the approach
 
# Function to print the pair that
# gives maximum nCr
def printMaxValPair(v, n):
 
    v.sort()
 
    # This gives the value of N in nCr
    N = v[n - 1]
 
    # Case 1 : When N is odd
    if N % 2 == 1:
        first_maxima = N // 2
        second_maxima = first_maxima + 1
        ans1, ans2 = 3 * (10 ** 18), 3 * (10 ** 18)
        from_left, from_right = -1, -1
        _from = -1
        for i in range(0, n):
            if v[i] > first_maxima:
                _from = i
                break
             
            else:
                diff = first_maxima - v[i]
                if diff < ans1:
                    ans1 = diff
                    from_left = v[i]
                 
        from_right = v[_from]
        diff1 = first_maxima - from_left
        diff2 = from_right - second_maxima
 
        if diff1 < diff2:
            print(N, from_left)
        else:
            print(N, from_right)
 
    # Case 2 : When N is even
    else:
        maxima = N // 2
        ans1 = 3 * (10 ** 18)
        R = -1
        for i in range(0, n - 1):
            diff = abs(v[i] - maxima)
            if diff < ans1:
                ans1 = diff
                R = v[i]
             
        print(N, R)
 
# Driver code
if __name__ == "__main__":
 
    v = [1, 1, 2, 3, 6, 1]
    n = len(v)
    printMaxValPair(v, n)
 
# This code is contributed by
# Rituraj Jain

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;
 
class GFG
{
 
// Function to print the pair that gives maximum nCr
static void printMaxValPair(List<long> v, int n)
{
    v.Sort();
 
    // This gives the value of N in nCr
    long N = v[(int)n - 1];
 
    // Case 1 : When N is odd
    if (N % 2 == 1)
    {
        long first_maxima = N / 2;
        long second_maxima = first_maxima + 1;
        long ans1 = (long) 3e18, ans2 = (long)3e18;
        long from_left = -1, from_right = -1;
        long from = -1;
        for (long i = 0; i < n; ++i)
        {
            if (v[(int)i] > first_maxima)
            {
                from = i;
                break;
            }
            else
            {
                long diff = first_maxima - v[(int)i];
                if (diff < ans1)
                {
                    ans1 = diff;
                    from_left = v[(int)i];
                }
            }
        }
        from_right = v[(int)from];
        long diff1 = first_maxima - from_left;
        long diff2 = from_right - second_maxima;
 
        if (diff1 < diff2)
            Console.WriteLine( N + " " + from_left);
        else
            Console.WriteLine( N + " " + from_right);
    }
 
    // Case 2 : When N is even
    else
    {
        long maxima = N / 2;
        long ans1 = (long)3e18;
        long R = -1;
        for (long i = 0; i < n - 1; ++i)
        {
            long diff = Math.Abs(v[(int)i] - maxima);
             
            if (diff < ans1)
            {
                ans1 = diff;
                R = v[(int)i];
            }
        }
        Console.WriteLine( N + " " + R);
    }
}
 
// Driver code
public static void Main(String []args)
{
    long []arr = { 1, 1, 2, 3, 6, 1};
    List<long> v = new List<long>( );
     
    for(int i = 0; i < arr.Length; i++)
    v.Add(arr[i]);
     
    int n = v.Count;
    printMaxValPair(v, n);
}
}
 
// This code contributed by Rajput-Ji

PHP

<?php
// PHP implementation of the approach
 
// Function to print the pair that
// gives maximum nCr
function printMaxValPair($v, $n)
{
    sort($v);
 
    // This gives the value of N in nCr
    $N = $v[$n - 1];
 
    // Case 1 : When N is odd
    if ($N % 2 == 1)
    {
        $first_maxima = $N / 2;
        $second_maxima = $first_maxima + 1;
        $ans1 = 3e18;
        $ans2 = 3e18;
        $from_left = -1;
        $from_right = -1;
        $from = -1;
        for ($i = 0; $i < $n; ++$i)
        {
            if ($v[$i] > $first_maxima)
            {
                $from = $i;
                break;
            }
            else
            {
                $diff = $first_maxima - $v[$i];
                if ($diff < $ans1)
                {
                    $ans1 = $diff;
                    $from_left = $v[$i];
                }
            }
        }
        $from_right = $v[$from];
        $diff1 = $first_maxima - $from_left;
        $diff2 = $from_right - $second_maxima;
 
        if ($diff1 < $diff2)
            echo $N . " " . $from_left;
        else
            echo $N . " " . $from_right;
    }
 
    // Case 2 : When N is even
    else
    {
        $maxima = $N / 2;
        $ans1 = 3e18;
        $R = -1;
        for ($i = 0; $i < $n - 1; ++$i)
        {
            $diff = abs($v[$i] - $maxima);
            if ($diff < $ans1)
            {
                $ans1 = $diff;
                $R = $v[$i];
            }
        }
        echo $N . " " . $R;
    }
}
 
// Driver code
$v = array( 1, 1, 2, 3, 6, 1 );
$n = count($v);
printMaxValPair($v, $n);
 
// This code is contributed by mits
?>

Javascript

<script>
  
// Javascript implementation of the approach
 
// Function to print the pair that gives maximum nCr
function printMaxValPair(v,  n)
{
    v.sort((a,b)=>a-b)
 
    // This gives the value of N in nCr
    var N = v[n - 1];
 
    // Case 1 : When N is odd
    if (N % 2 == 1) {
        var first_maxima = N / 2;
        var second_maxima = first_maxima + 1;
        var ans1 = 3000000000000000000, ans2 = 3000000000000000000;
        var from_left = -1, from_right = -1;
        var from = -1;
        for (var i = 0; i < n; ++i) {
            if (v[i] > first_maxima) {
                from = i;
                break;
            }
            else {
                var diff = first_maxima - v[i];
                if (diff < ans1) {
                    ans1 = diff;
                    from_left = v[i];
                }
            }
        }
        from_right = v[from];
        var diff1 = first_maxima - from_left;
        var diff2 = from_right - second_maxima;
 
        if (diff1 < diff2)
            document.write( N + " " + from_left);
        else
            document.write( N + " " + from_right);
    }
 
    // Case 2 : When N is even
    else {
        var maxima = parseInt(N / 2);
        var ans1 = 3000000000000000000;
        var R = -1;
        for (var i = 0; i < n - 1; ++i) {
            var diff = Math.abs(v[i] - maxima);
            if (diff < ans1) {
                ans1 = diff;
                R = v[i];
            }
        }
        document.write( N + " " + R);
    }
}
 
// Driver code
var v = [1, 1, 2, 3, 6, 1 ];
var n = v.length;
printMaxValPair(v, n);
 
// This code is contributed by famously.
</script>

Output:

6 3

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