Related Articles
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 ``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

 ` 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

 ``

Output :

`6`

My Personal Notes arrow_drop_up