The problem involves a triangle $\triangle ABC$, where points are placed on the sides of the triangle, and we are tasked with finding how many triangles can be formed using these points with certain restrictions.

Let's break down the key information:

• Points on sides:

  • There are 7 points on side AB,
  • 7 points on side BC, and
  • 8 points on side AC.

• Triangle restrictions: We need to find the number of triangles that can be formed, with the condition that at most one vertex lies on side AB.

Solution strategy:

1. Total number of points:

   • Total points on sides AB, BC, and AC are $7 + 7 + 8 = 22$.

2. Total number of triangles without restrictions:

   • Any triangle is formed by choosing 3 points from the 22 available points, which is computed by: $\binom{22}{3} = \frac{22 \times 21 \times 20}{3 \times 2 \times 1} = 1540$

3. Removing triangles where all vertices lie on one side:

   • We cannot form a valid triangle using three collinear points, so we need to exclude triangles where all three vertices are on one side:

     • AB: There are 7 points, so the number of triangles using only points from AB is $\binom{7}{3} = 35$.
     • BC: Similarly, $\binom{7}{3} = 35$ for BC.
     • AC: For AC, $\binom{8}{3} = 56$.

   • Total number of invalid triangles (on one side) = $35 + 35 + 56 = 126$.

4. Valid triangles without restrictions: $\text{Valid triangles} = 1540 - 126 = 1414$

5. Applying the condition of "at most one vertex on AB":

   • We need to exclude triangles where two or more vertices lie on side AB:

     • Two vertices on AB, one on BC: $\binom{7}{2} \times \binom{7}{1} = 21 \times 7 = 147$.
     • Two vertices on AB, one on AC: $\binom{7}{2} \times \binom{8}{1} = 21 \times 8 = 168$.

   • Total triangles with two or more vertices on AB = $147 + 168 = 315$.

6. Final number of valid triangles: $\text{Valid triangles} = 1414 - 315 = 1099$

Thus, the number of triangles that can be formed with at most one vertex on side AB is 1099.

Would you like more details or have any questions?

Here are 5 follow-up questions related to this problem:

1. How do we generalize the approach if the number of points on each side is different?
2. What would change if the restriction was "at least two vertices on AB"?
3. Can you explain why choosing all vertices from one side doesn’t form a triangle?
4. How do we calculate the number of valid triangles if the number of points on BC or AC changes?
5. How would the solution differ if the points were allowed to coincide with the vertices of the triangle?

Tip: Understanding combinations is key to solving problems where we need to choose subsets, especially in geometric arrangements like this!