Question

a lattice point is a point with integer coordinates such as 2, 3. in how many ways can we pick 3 lattice points such that both coordinates of each point are positive integers less than 5, and the three points form a triangle?

Answer 1

Triangles are formed by choosing 3 non-collinear lattice points with coordinates less than 5. The formula C(16, 3) minus the straight-line combinations gives us the answer.

Step-by-step explanation:

We need to count the number of triangles that can be formed by choosing three lattice points where both coordinates are positive integers less than 5. There are 4x4 = 16 such lattice points in total. To form a triangle, we need to choose any 3 points that don't all lie on the same straight line. We can use the combination formula C(n, k) = n! / [k!(n-k)!] where n is the total number of points and k is the number of points to choose.

Using the formula C(16, 3), we can calculate the number of different combinations of three points, which gives us 560 possible combinations. However, this includes combinations where the points fall in a straight line. There are 4 horizontal and 4 vertical lines with 4 points each and 2 main diagonals with 4 points. Each line can form 4 choose 2 = 6 lines, so we subtract 6 lines times 10 (4+4+2). Thus, we subtract 60 from 560.

Additionally, there will be combinations on the smaller diagonals that also fall in a straight line, and we also need to subtract those. This includes diagonals within 3x3, 2x2 and 3x2 rectangles formed within the grid. After identifying and subtracting all these combinations, we derive the final number of triangles that can be formed.