Maximize the value of the given expression

Given three non-zero integers a, b and c. The task is to find the maximum value possible by putting addition and multiplication signs between them in any order. 

Note: Rearrangement of integers is allowed but addition and multiplication sign must be used once. Braces can also be placed between equations as per your need.

Examples:

Input: a = 2, b = 1, c = 4 
Output: 12 
(1 + 2) * 4 = 3 * 4 = 12

Input: a = 2, b = 2, c = 2 
Output: 8 
(2 + 2) * 2 = 4 * 2 = 8 

Approach: To solve this problem one can opt the method of generating all the possibilities and calculate them to get the maximum value but this approach is not efficient. Take the advantage of given conditions that integers may got rearranged and mandatory use of each mathematical sign (+, *). There are total of four cases to solve which are listed below:  

1. All three integers are non-negative: For this simply add two smaller one and multiply their result by largest integer.
2. One integer is negative and rest two positive : Multiply the both positive integer and add their result to negative integer.
3. Two integers are negative and one is positive: As the product of two negative numbers is positive multiply both negative integers and then add their result to positive integer.
4. All three are negative integers: add the two largest integers and multiply them to smallest one. case 3-: (sum – smallest) * smallest

Below is the implementation of the above approach:  

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to return the maximum result
int maximumResult(int a, int b, int c)
{

    // To store the count of negative integers
    int countOfNegative = 0;

    // Sum of all the three integers
    int sum = a + b + c;

    // Product of all the three integers
    int product = a * b * c;

    // To store the smallest and the largest
    // among all the three integers
    int largest = max(a,max(b,c));
    int smallest = min(a,min(b,c) );


    // Calculate the count of negative integers
    if (a < 0)
        countOfNegative++;
    if (b < 0)
        countOfNegative++;
    if (c < 0)
        countOfNegative++;

    // Depending upon count of negatives
    switch (countOfNegative) {

    // When all three are positive integers
    case 0:
        return (sum - largest) * largest;

    // For single negative integer
    case 1:
        return (product / smallest) + smallest;

    // For two negative integers
    case 2:
        return (product / largest) + largest;

    // For three negative integers
    case 3:
        return (sum - smallest) * smallest;
    }
}

// Driver Code
int main()
{
    int a=-2,b=-1,c=-4;
    cout << maximumResult(a, b, c);

    return 0;
}
// This code contributed by Nikhil

Java

// Java implementation of the approach
class GFG
{

// Function to return the maximum result
static int maximumResult(int a, int b, int c)
{

    // To store the count of negative integers
    int countOfNegative = 0;

    // Sum of all the three integers
    int sum = a + b + c;

    // Product of all the three integers
    int product = a * b * c;

    // To store the smallest and the largest
    // among all the three integers
    int largest = (a > b) ? ((a > c) ? a : c) :
                            ((b > c) ? b : c);
    int smallest= (a<b)?((a<c)? a : c):((b<c) ? b : c);
    // Calculate the count of negative integers
    if (a < 0)
        countOfNegative++;
    if (b < 0)
        countOfNegative++;
    if (c < 0)
        countOfNegative++;

    // Depending upon count of negatives
    switch (countOfNegative)
    {

        // When all three are positive integers
        case 0:
            return (sum - largest) * largest;

        // For single negative integer
        case 1:
            return (product / smallest) + smallest;

        // For two negative integers
        case 2:
            return (product / largest) + largest;

        // For three negative integers
        case 3:
            return (sum - smallest) * smallest;
    }
    return -1;
}

// Driver Code
public static void main(String[] args)
{
    int a=-2,b=-1,c=-4;
    System.out.print(maximumResult(a, b, c));
}
}

// This code contributed by Nikhil

Python3

# Python3 implementation of the approach

# Function to return the maximum result
# Python3 implementation of the approach

# Function to return the maximum result
def maximumResult(a, b, c):

    # To store the count of negative integers
    countOfNegative = 0

    # Sum of all the three integers
    Sum = a + b + c

    # Product of all the three integers
    product = a * b * c

    # To store the smallest and the
    # largest among all the three integers
    largest = max(a, b, c)
    smallest = min(a, b, c)

    # Calculate the count of negative integers
    if a < 0:
        countOfNegative += 1
    if b < 0:
        countOfNegative += 1
    if c < 0:
        countOfNegative += 1

    # When all three are positive integers
    if countOfNegative == 0:
        return (Sum - largest) * largest

    # For single negative integer
    elif countOfNegative == 1:
        return (product // smallest) + smallest

    # For two negative integers
    elif countOfNegative == 2:
        return (product // largest) + largest

    # For three negative integers
    elif countOfNegative == 3:
        return (Sum - smallest) * smallest

# Driver Code
if __name__ == "__main__":

    a, b, c = -2, -1, -4
    print(maximumResult(a, b, c))

C#

// C# implementation of the approach
using System;

class GFG
{

// Function to return the maximum result
static int maximumResult(int a, int b, int c)
{

    // To store the count of negative integers
    int countOfNegative = 0;

    // Sum of all the three integers
    int sum = a + b + c;

    // Product of all the three integers
    int product = a * b * c;

    // To store the smallest and the largest
    // among all the three integers
    int largest = (a > b) ? ((a > c) ? a : c) :
                            ((b > c) ? b : c);
    int smallest=(a<b)?((a<c)? a : c):((b<c) ? b : c);

    // Calculate the count of negative integers
    if (a < 0)
        countOfNegative++;
    if (b < 0)
        countOfNegative++;
    if (c < 0)
        countOfNegative++;

    // Depending upon count of negatives
    switch (countOfNegative)
    {

        // When all three are positive integers
        case 0:
            return (sum - largest) * largest;

        // For single negative integer
        case 1:
            return (product / smallest) + smallest;

        // For two negative integers
        case 2:
            return (product / largest) + largest;

        // For three negative integers
        case 3:
            return (sum - smallest) * smallest;
    }
    return -1;
}

// Driver Code
static void Main()
{
    int a = -2, b = -1, c = -4;
    Console.WriteLine(maximumResult(a, b, c));
}
}

// This code is contributed by mits & Nikhil

PHP

<?php
// PHP implementation of the approach

// Function to return the maximum result
function maximumResult($a, $b, $c)
{

    // To store the count of
    // negative integers
    $countOfNegative = 0;

    // Sum of all the three integers
    $sum = $a + $b + $c;

    // Product of all the three integers
    $product = $a * $b * $c;

    // To store the smallest and the largest
    // among all the three integers
    $largest = max($a ,$b, $c);
    $smallest = min($a ,$b, $c);


    // Calculate the count of negative integers
    if ($a < 0)
        $countOfNegative++;
    if ($b < 0)
        $countOfNegative++;
    if ($c < 0)
        $countOfNegative++;

    // Depending upon count of negatives
    switch ($countOfNegative)
    {

        // When all three are positive integers
        case 0:
            return ($sum - $largest) *
                           $largest;

        // For single negative integer
        case 1:
            return ($product / $smallest) +
                               $smallest;

        // For two negative integers
        case 2:
            return ($product / $largest) +
                               $largest;

        // For three negative integers
        case 3:
             return ($sum - $smallest) *
                           $smallest;
    }
}

// Driver Code
$a = -2;
$b = -1;
$c = -4;
echo maximumResult($a, $b, $c);

// This code is contributed by ihritik
?>

Javascript

<script>
// JavaScript implementation of the approach

// Function to return the maximum result
function maximumResult(a, b, c)
{

    // To store the count of negative integers
    let countOfNegative = 0;

    // Sum of all the three integers
    let sum = a + b + c;

    // Product of all the three integers
    let product = a * b * c;

    // To store the smallest and the largest
    // among all the three integers
    let largest = Math.max(a,Math.max(b,c));
    let smallest = Math.min(a,Math.min(b,c) );

    // Calculate the count of negative integers
    if (a < 0)
        countOfNegative++;
    if (b < 0)
        countOfNegative++;
    if (c < 0)
        countOfNegative++;

    // Depending upon count of negatives
    switch (countOfNegative) {

    // When all three are positive integers
    case 0:
        return (sum - largest) * largest;

    // For single negative integer
    case 1:
        return (product / smallest) + smallest;

    // For two negative integers
    case 2:
        return (product / largest) + largest;

    // For three negative integers
    case 3:
        return (sum - smallest) * smallest;
    }
}

// Driver Code

    let a = -2, b = -1, c = -4;
    document.write(maximumResult(a, b, c));

// This code is contributed by Surbhi Tyagi.
</script>

Output:

12

Time Complexity: O(1)

Auxiliary Space: O(1)