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Count of triangles with total n points with m collinear
  • Last Updated : 09 Apr, 2021

There are ‘n’ points in a plane, out of which ‘m’ points are co-linear. Find the number of triangles formed by the points as vertices ?
Examples : 
 

Input :  n = 5, m = 4
Output : 6
Out of five points, four points are 
collinear, we can make 6 triangles. We
can choose any 2 points from 4 collinear
points and use the single point as 3rd
point. So total count is 4C2 = 6

Input :  n = 10, m = 4
Output : 116

 

 

Number of triangles = nC3mC3
How does this formula work? 
Consider the second example above. There are 10 points, out of which 4 collinear. A triangle will be formed by any three of these ten points. Thus forming a triangle amounts to selecting any three of the 10 points. Three points can be selected out of the 10 points in nC3 ways. 
Number of triangles formed by 10 points when no 3 of them are co-linear = 10C3……(i) 
Similarly, the number of triangles formed by 4 points when no 3 of them are co-linear = 4C3……..(ii)
Since triangle formed by these 4 points are not valid, required number of triangles formed = 10C34C3 = 120 – 4 = 116 

 

C++




// CPP program to count number of triangles
// with n total points, out of which m are
// collinear.
#include <bits/stdc++.h>
using namespace std;
 
// Returns value of binomial coefficient
// Code taken from https://goo.gl/vhy4jp
int nCk(int n, int k)
{
    int C[k+1];
    memset(C, 0, sizeof(C));
 
    C[0] = 1;  // nC0 is 1
 
    for (int i = 1; i <= n; i++)
    {
        // Compute next row of pascal triangle
        // using the previous row
        for (int j = min(i, k); j > 0; j--)
            C[j] = C[j] + C[j-1];
    }
    return C[k];
}
 
/* function to calculate number of triangle
   can be formed */
int counTriangles(int n,int m)
{
    return (nCk(n, 3) - nCk(m, 3));
}
 
/* driver function*/
int main()
{
    int n = 5, m = 4;
    cout << counTriangles(n, m);
    return 0;
}

Java




//Java program to count number of triangles
// with n total points, out of which m are
// collinear.
import java.io.*;
import java.util.*;
 
class GFG {
 
// Returns value of binomial coefficient
// Code taken from https://goo.gl/vhy4jp
static int nCk(int n, int k)
{
    int[] C=new int[k+1];
    for (int i=0;i<=k;i++)
    C[i]=0;
     
    C[0] = 1; // nC0 is 1
 
    for (int i = 1; i <= n; i++)
    {
        // Compute next row of pascal triangle
        // using the previous row
        for (int j = Math.min(i, k); j > 0; j--)
            C[j] = C[j] + C[j-1];
    }
    return C[k];
}
 
/* function to calculate number of triangle
can be formed */
static int counTriangles(int n,int m)
{
    return (nCk(n, 3) - nCk(m, 3));
}
 
    public static void main (String[] args) {
      int n = 5, m = 4;
      System.out.println(counTriangles(n, m));
 
    }
}
 
//This code is contributed by Gitanjali.

Python3




# python program to count number of triangles
# with n total points, out of which m are
# collinear.
import math
  
# Returns value of binomial coefficient
# Code taken from https://goo.gl / vhy4jp
def nCk(n, k):
    C = [0 for i in range(0, k + 2)]
  
    C[0] = 1; # nC0 is 1
    for i in range(0, n + 1):
      
        # Compute next row of pascal triangle
        # using the previous row
        for j in range(min(i, k), 0, -1):
            C[j] = C[j] + C[j-1]
      
    return C[k]
  
# function to calculate number of triangle
# can be formed
def counTriangles(n, m):
    return (nCk(n, 3) - nCk(m, 3))
  
# driver code
n = 5
m = 4
print (counTriangles(n, m))
  
# This code is contributed by Gitanjali

C#




//C# program to count number of triangles
// with n total points, out of which m are
// collinear.
using System;
 
class GFG {
 
    // Returns value of binomial coefficient
    // Code taken from https://goo.gl/vhy4jp
    static int nCk(int n, int k)
    {
        int[] C=new int[k+1];
        for (int i = 0; i <= k; i++)
        C[i] = 0;
         
        // nC0 is 1
        C[0] = 1;
     
        for (int i = 1; i <= n; i++)
        {
            // Compute next row of pascal triangle
            // using the previous row
            for (int j = Math.Min(i, k); j > 0; j--)
                C[j] = C[j] + C[j - 1];
        }
        return C[k];
    }
     
    /* function to calculate number of triangle
    can be formed */
    static int counTriangles(int n,int m)
    {
        return (nCk(n, 3) - nCk(m, 3));
    }
     
    // Driver code
    public static void Main ()
    {
        int n = 5, m = 4;
        Console.WriteLine(counTriangles(n, m));
 
    }
}
 
// This code is contributed by vt_m.

PHP




<?php
// PHP program to count number
// of triangles with n total
// points, out of which m are collinear.
 
// Returns value of binomial coefficient
// Code taken from https://goo.gl/vhy4jp
function nCk($n, $k)
{
    for ($i = 0; $i <= $k; $i++)
    $C[$i] = 0;
 
    $C[0] = 1; // nC0 is 1
 
    for ($i = 1; $i <= $n; $i++)
    {
        // Compute next row of pascal
        // triangle using the previous row
        for ($j = min($i, $k); $j > 0; $j--)
            $C[$j] = $C[$j] + $C[$j - 1];
    }
    return $C[$k];
}
 
/* function to calculate number
of triangles that can be formed */
function counTriangles($n, $m)
{
    return (nCk($n, 3) - nCk($m, 3));
}
 
// Driver Code
$n = 5;
$m = 4;
echo counTriangles($n, $m);
return 0;
 
// This code is contributed by ChitraNayal
?>

Javascript




<script>
    // Javascript program to count number of triangles
    // with n total points, out of which m are
    // collinear.
     
    // Returns value of binomial coefficient
    // Code taken from https://goo.gl/vhy4jp
    function nCk(n, k)
    {
        let C = new Array(k+1);
        C.fill(0);
 
        C[0] = 1;  // nC0 is 1
 
        for (let i = 1; i <= n; i++)
        {
            // Compute next row of pascal triangle
            // using the previous row
            for (let j = Math.min(i, k); j > 0; j--)
                C[j] = C[j] + C[j-1];
        }
        return C[k];
    }
 
    /* function to calculate number of triangle
       can be formed */
    function counTriangles(n, m)
    {
        return (nCk(n, 3) - nCk(m, 3));
    }
     
    let n = 5, m = 4;
    document.write(counTriangles(n, m));
     
    // This code is contributed by divyesh072019.
</script>

Output : 
 

6

 




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