# Count ways to form minimum product triplets

Given an array of positive integers. We need to find how many triples of indices (i, j, k) (i < j < k), such that a[i] * a[j] * a[k] is minimum possible.

Examples:Input : 5 1 3 2 3 4 Output : 2 The triplets are (1, 3, 2) and (1, 2, 3) Input : 5 2 2 2 2 2 Output : 5 In this example we choose three 2s out of five, and the number of ways to choose them is^{5}C_{3}. Input : 6 1 3 3 1 3 2 Output : 1 There is only one way (1, 1, 2).

Following cases arise in this problem.

- All three minimum elements are same. For example {1, 1, 1, 1, 2, 3, 4}. The solution for such cases is
^{n}C_{3}.- Two elements are same. For example {1, 2, 2, 2, 3} or {1, 1, 2, 2}. In this case, count of occurrences of first (or minimum element) cannot be more than 2. If minimum element appears two times, then answer is count of second element (We get to choose only 1 from all occurrences of second element. If minimum element appears once, the count is
^{n}C_{2}.- All three elements are distinct. For example {1, 2, 3, 3, 5}. In this case, answer is count of occurrences of third element (or
^{n}C_{1}).

We first sort the array in increasing order. Then count the frequency of 3 element of 3rd element from starting. Let the frequency be ‘count’. Following cases arise.

- If 3rd element is equal to the first element, no. of triples will be (count-2)*(count-1)*(count)/6, where count is the frequency of 3rd element.
- If 3rd element is equal to 2nd element, no. of triples will be (count-1)*(count)/2. Otherwise no. of triples will be value of count.

**Implementation:**

## C++

`// CPP program to count number of ways we can` `// form triplets with minimum product.` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// function to calculate number of triples` `long` `long` `noOfTriples(` `long` `long` `arr[], ` `int` `n)` `{` ` ` `// Sort the array` ` ` `sort(arr, arr + n);` ` ` `// Count occurrences of third element` ` ` `long` `long` `count = 0;` ` ` `for` `(` `long` `long` `i = 0; i < n; i++)` ` ` `if` `(arr[i] == arr[2])` ` ` `count++;` ` ` ` ` `// If all three elements are same (minimum` ` ` `// element appears at least 3 times). Answer` ` ` `// is nC3.` ` ` `if` `(arr[0] == arr[2])` ` ` `return` `(count - 2) * (count - 1) * (count) / 6;` ` ` `// If minimum element appears once. ` ` ` `// Answer is nC2.` ` ` `else` `if` `(arr[1] == arr[2])` ` ` `return` `(count - 1) * (count) / 2;` ` ` ` ` `// Minimum two elements are distinct.` ` ` `// Answer is nC1.` ` ` `return` `count;` `}` `// Driver code` `int` `main()` `{` ` ` `long` `long` `arr[] = { 1, 3, 3, 4 };` ` ` `int` `n = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `cout << noOfTriples(arr, n);` ` ` `return` `0;` `}` |

## Java

`// Java program to count number of ways we can` `// form triplets with minimum product.` `import` `java.util.Arrays;` `class` `GFG {` ` ` ` ` `// function to calculate number of triples` ` ` `static` `long` `noOfTriples(` `long` `arr[], ` `int` `n)` ` ` `{` ` ` ` ` `// Sort the array` ` ` `Arrays.sort(arr);` ` ` ` ` `// Count occurrences of third element` ` ` `long` `count = ` `0` `;` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++)` ` ` `if` `(arr[i] == arr[` `2` `])` ` ` `count++;` ` ` ` ` `// If all three elements are same (minimum` ` ` `// element appears at least 3 times). Answer` ` ` `// is nC3.` ` ` `if` `(arr[` `0` `] == arr[` `2` `])` ` ` `return` `(count - ` `2` `) * (count - ` `1` `) *` ` ` `(count) / ` `6` `;` ` ` ` ` `// If minimum element appears once.` ` ` `// Answer is nC2.` ` ` `else` `if` `(arr[` `1` `] == arr[` `2` `])` ` ` `return` `(count - ` `1` `) * (count) / ` `2` `;` ` ` ` ` `// Minimum two elements are distinct.` ` ` `// Answer is nC1.` ` ` `return` `count;` ` ` `}` ` ` ` ` `//driver code` ` ` `public` `static` `void` `main(String arg[])` ` ` `{` ` ` ` ` `long` `arr[] = { ` `1` `, ` `3` `, ` `3` `, ` `4` `};` ` ` `int` `n = arr.length;` ` ` ` ` `System.out.print(noOfTriples(arr, n));` ` ` `}` `}` `// This code is contributed by Anant Agarwal.` |

## Python3

`# Python3 program to count number` `# of ways we can form triplets` `# with minimum product.` `# function to calculate number of triples` `def` `noOfTriples (arr, n):` ` ` `# Sort the array` ` ` `arr.sort()` ` ` ` ` `# Count occurrences of third element` ` ` `count ` `=` `0` ` ` `for` `i ` `in` `range` `(n):` ` ` `if` `arr[i] ` `=` `=` `arr[` `2` `]:` ` ` `count` `+` `=` `1` ` ` ` ` `# If all three elements are same` ` ` `# (minimum element appears at l` ` ` `# east 3 times). Answer is nC3.` ` ` `if` `arr[` `0` `] ` `=` `=` `arr[` `2` `]:` ` ` `return` `(count ` `-` `2` `) ` `*` `(count ` `-` `1` `) ` `*` `(count) ` `/` `6` ` ` ` ` `# If minimum element appears once.` ` ` `# Answer is nC2.` ` ` `elif` `arr[` `1` `] ` `=` `=` `arr[` `2` `]:` ` ` `return` `(count ` `-` `1` `) ` `*` `(count) ` `/` `2` ` ` ` ` `# Minimum two elements are distinct.` ` ` `# Answer is nC1.` ` ` `return` `count` ` ` `# Driver code` `arr ` `=` `[` `1` `, ` `3` `, ` `3` `, ` `4` `]` `n ` `=` `len` `(arr)` `print` `(noOfTriples(arr, n))` `# This code is contributed by "Abhishek Sharma 44"` |

## C#

`// C# program to count number of ways` `// we can form triplets with minimum product.` `using` `System;` `class` `GFG {` ` ` `// function to calculate number of triples` `static` `long` `noOfTriples(` `long` `[]arr, ` `int` `n)` `{` ` ` `// Sort the array` ` ` `Array.Sort(arr);` ` ` ` ` `// Count occurrences of third element` ` ` `long` `count = 0;` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `if` `(arr[i] == arr[2])` ` ` `count++;` ` ` ` ` `// If all three elements are same (minimum` ` ` `// element appears at least 3 times). Answer` ` ` `// is nC3.` ` ` `if` `(arr[0] == arr[2])` ` ` `return` `(count - 2) * (count - 1) * (count) / 6;` ` ` ` ` `// If minimum element appears once. ` ` ` `// Answer is nC2.` ` ` `else` `if` `(arr[1] == arr[2])` ` ` `return` `(count - 1) * (count) / 2;` ` ` ` ` `// Minimum two elements are distinct.` ` ` `// Answer is nC1.` ` ` `return` `count;` `}` `//driver code` `public` `static` `void` `Main()` `{` ` ` `long` `[]arr = { 1, 3, 3, 4 };` ` ` `int` `n = arr.Length;` ` ` `Console.Write(noOfTriples(arr, n));` `}` `}` `//This code is contributed by Anant Agarwal.` |

## PHP

`<?php` `// PHP program to count number of ways` `// we can form triplets with minimum` `// product.` `// function to calculate number of` `// triples` `function` `noOfTriples( ` `$arr` `, ` `$n` `)` `{` ` ` `// Sort the array` ` ` `sort(` `$arr` `);` ` ` `// Count occurrences of third element` ` ` `$count` `= 0;` ` ` `for` `( ` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++)` ` ` `if` `(` `$arr` `[` `$i` `] == ` `$arr` `[2])` ` ` `$count` `++;` ` ` ` ` `// If all three elements are same` ` ` `// (minimum element appears at least` ` ` `// 3 times). Answer is nC3.` ` ` `if` `(` `$arr` `[0] == ` `$arr` `[2])` ` ` `return` `(` `$count` `- 2) * (` `$count` `- 1) ` ` ` `* (` `$count` `) / 6;` ` ` `// If minimum element appears once.` ` ` `// Answer is nC2.` ` ` `else` `if` `(` `$arr` `[1] == ` `$arr` `[2])` ` ` `return` `(` `$count` `- 1) * (` `$count` `) / 2;` ` ` `// Minimum two elements are distinct.` ` ` `// Answer is nC1.` ` ` `return` `$count` `;` `}` `// Driver code` ` ` `$arr` `= ` `array` `( 1, 3, 3, 4 );` ` ` `$n` `= ` `count` `(` `$arr` `);` ` ` `echo` `noOfTriples(` `$arr` `, ` `$n` `);` `// This code is contributed by anuj_67.` `?>` |

## Javascript

`<script>` ` ` `// Javascript program to count number of ways we can` ` ` `// form triplets with minimum product.` ` ` ` ` `// function to calculate number of triples` ` ` `function` `noOfTriples(arr, n)` ` ` `{` ` ` `// Sort the array` ` ` `arr.sort();` ` ` `// Count occurrences of third element` ` ` `let count = 0;` ` ` `for` `(let i = 0; i < n; i++)` ` ` `if` `(arr[i] == arr[2])` ` ` `count++;` ` ` `// If all three elements are same (minimum` ` ` `// element appears at least 3 times). Answer` ` ` `// is nC3.` ` ` `if` `(arr[0] == arr[2])` ` ` `return` `(count - 2) * (count - 1) * (count) / 6;` ` ` `// If minimum element appears once. ` ` ` `// Answer is nC2.` ` ` `else` `if` `(arr[1] == arr[2])` ` ` `return` `(count - 1) * (count) / 2;` ` ` `// Minimum two elements are distinct.` ` ` `// Answer is nC1.` ` ` `return` `count;` ` ` `}` ` ` `let arr = [ 1, 3, 3, 4 ];` ` ` `let n = arr.length;` ` ` `document.write(noOfTriples(arr, n));` ` ` ` ` `// This code is contributed by mukesh07.` `</script>` |

**Output**

1

**Time Complexity: O(n Log n) ****Auxiliary Space: O(1)**

The solution can be optimized by first finding minimum element and its frequency and if frequency is less than 3, then finding second minimum and its frequency. If overall frequency is less than 3, then finding third minimum and its frequency. Time complexity of this optimized solution would be O(n)

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