There are 12 points in a plane of which 5 are collinear. The number of triangles is

To determine the number of triangles that can be formed with 12 points in a plane, where 5 of those points are collinear, we need to break down the problem.

1. ### Break Down the Problem

• Total points = 12
• Points that are collinear = 5
• Number of points not collinear = 12 - 5 = 7

A triangle cannot be formed using 3 collinear points. So, we will calculate the total number of triangles that can be formed and then subtract those formed by the collinear points.

2. ### Relevant Concepts

• The number of ways to choose 3 points from n points is given by the combination formula:

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

where $n$ is the total number of points and $r$ is the number of points to choose.

3. ### Analysis and Detail

• Calculate the total number of triangles that can be formed using all 12 points:

$\text{Total triangles} = \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220$

• Calculate the number of triangles that can be formed using the 5 collinear points (which do not count as valid triangles):

$\text{Collinear triangles} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10$

• Subtract the collinear triangles from the total triangles to get the valid triangles:

$\text{Valid triangles} = \text{Total triangles} - \text{Collinear triangles} = 220 - 10 = 210$

4. ### Verify and Summarize

The calculations are verified as follows:

• Total valid triangles calculated is 210, after subtracting the invalid combinations of collinear points.

Final Answer

The number of triangles that can be formed is 210.