Let's break down how to calculate the number of triangles that can be formed from a given set of points, especially when some of those points lie on the same straight line (are collinear).

We are given:

• Total number of points in a plane: 12
• Number of points that lie on the same straight line (collinear points): 7

A triangle is formed by choosing 3 points that are not collinear. If we choose 3 points that are collinear, they will lie on a straight line and cannot form a triangle.

Calculating Total Ways to Choose 3 Points

First, let's find the total number of ways to choose any 3 points from the 12 points, without considering if they are collinear or not. This can be done using the combination formula \(\binom{n}{k}\), where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.

Here, $n = 12$ (total points) and $k = 3$ (points needed for a triangle).

The total number of ways to choose 3 points from 12 is:

\(\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!}\)

Let's calculate this value:

\(\binom{12}{3} = \frac{12 \times 11 \times 10 \times 9!}{ (3 \times 2 \times 1) \times 9!} = \frac{12 \times 11 \times 10}{6}\)

\(\binom{12}{3} = 2 \times 11 \times 10 = 220\)

So, there are 220 ways to choose any 3 points from the 12 points.

Excluding Collinear Point Combinations

Next, we need to find the number of ways to choose 3 points specifically from the set of 7 collinear points. Any combination of 3 points chosen from these 7 will be collinear and will not form a triangle.

Using the combination formula again, with $n = 7$ (collinear points) and $k = 3$ (points to choose):

The number of ways to choose 3 points from the 7 collinear points is:

\(\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\)

Let's calculate this value:

\(\binom{7}{3} = \frac{7 \times 6 \times 5 \times 4!}{(3 \times 2 \times 1) \times 4!} = \frac{7 \times 6 \times 5}{6}\)

\(\binom{7}{3} = 7 \times 5 = 35\)

So, there are 35 ways to choose 3 points that are collinear and therefore cannot form a triangle.

Calculating the Number of Triangles

The number of triangles that can be formed is the total number of ways to choose 3 points minus the number of ways to choose 3 collinear points (which do not form triangles).

Number of Triangles = (Total ways to choose 3 points) - (Ways to choose 3 collinear points)

Number of Triangles = \(\binom{12}{3} - \binom{7}{3}\)

Number of Triangles = $220 - 35$

Number of Triangles = $185$

Therefore, 185 triangles can be formed by choosing vertices from the set of 12 points, seven of which are collinear.

The final answer is 185.

Revision Table: Triangle Formation from Points

Additional Information: Combinations and Geometry

This problem combines concepts from combinatorics (counting principles) and basic geometry.

• Combinations: A combination is a selection of items from a collection, such that the order of selection does not matter. The formula \(\binom{n}{k}\) is used when we want to find the number of ways to choose $k$ items from a set of $n$ distinct items.
• Collinear Points: Points that lie on a single straight line. Any three or more points on the same line are collinear. A fundamental geometric principle is that three points are needed to define a triangle, and these three points must not be collinear.
• Problem Strategy: Problems involving forming geometric shapes (like triangles or lines) from a set of points, where some points might be collinear, often require calculating the total possible combinations and then subtracting the combinations that violate the geometric condition (e.g., 3 collinear points for a triangle).