Find Number of Triangles Formed from Points on Three Lines Using C++

We are given several points present in 3 lines now; we are required to find how many triangles these points can form, for example

Input: m = 3, n = 4, k = 5
Output: 205

Input: m = 2, n = 2, k = 1
Output: 10

We will apply some combinatorics to this question and make up some formulas to solve this problem.

Approach to find The Solution

In this approach, we will devise a formula by applying combinatorics to the current situations, and this formula will give us our results.

C++ Code for the Above Approach

Here is the C++ syntax which we can use as an input to solve the given problem −

Example

#include <bits/stdc++.h>

#define MOD 1000000007

using namespace std;

long long fact(long long n) {
   if(n <= 1)
   return 1;
   return ((n % MOD) * (fact(n-1) % MOD)) % MOD;
}
long long comb(int n, int r) {
   return (((fact(n)) % MOD) / ((fact(r) % MOD) * (fact(n-r) % MOD)) % MOD);
}

int main() {
   int n = 3;
   int m = 4;
   int r = 5;
   long long linen = comb(n, 3); // the combination of n with 3.
   long long linem = comb(m, 3); // the combination of m with 3.
   long long liner = comb(r, 3); //the combination of r with 3.
   long long answer = comb(n + m + r, 3); // all possible comb of n, m , r with 3.
   answer -= (linen + linem + liner);
   cout << answer << "\n";
   return 0;
}

Output

205

Explanation of the above code

In this approach, we find all the possible combinations of n+m+r with three, i.e., comb(n+m+r, 3). Now, as you know, the conditions for 3 points to be a triangle is that they shouldn't be collinear, so we find all the possible collinear points that are the sum of the combination of n, m, r with three, and when we subtract this sum with the variety of n+m+r with three we get the answer, and we print it.

Conclusion

This article discussed how many triangles can be formed from a set of points on three lines by applying some combinatorics. We also learned the C++ program for this problem and the complete approach (Normal) by which we solved this problem. We can write the same program in other languages such as C, java, python, and other languages. We hope you find this article helpful.