How many triangles may be formed by joining any three of the nine points when
five of them are collinear?

Answer

To solve the problem of how many triangles can be formed by joining any three of the nine points when five of them are collinear, we can follow these steps:

### Step 1: Understand the problem
We have a total of 9 points, out of which 5 points are collinear. We need to find out how many triangles can be formed by selecting any three points from these 9 points.

### Step 2: Calculate the total combinations of points
First, we calculate the total number of ways to choose any 3 points from the 9 points without considering the collinearity. This can be calculated using the combination formula:

\[ \text{Total combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Here, \( n = 9 \) and \( r = 3 \):

\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!} \]

### Step 3: Simplify the combination
Now, we simplify the expression:

\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \]

So, there are 84 ways to choose any 3 points from the 9 points.

### Step 4: Exclude the collinear combinations
Since 5 points are collinear, any triangle formed by selecting 3 points from these 5 points will not be a valid triangle. We need to calculate the number of ways to choose 3 points from these 5 collinear points:

\[ \text{Collinear combinations} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the valid triangles
Now, we subtract the number of invalid triangles (formed by collinear points) from the total combinations:

\[ \text{Valid triangles} = \text{Total combinations} - \text{Collinear combinations} = 84 - 10 = 74 \]

### Final Answer
Thus, the total number of triangles that can be formed by joining any three of the nine points, considering that five of them are collinear, is **74**.

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