IGDTUW 2026 — Mathematics PYQ

Step 1: Concept and Formula

• To form a single triangle, we need to select exactly $3$ non-collinear points.

• The number of ways to choose $r$ objects out of $n$ distinct objects is given by the combination formula:

  $${}^n{C}_r=\frac{n!}{r!(n-r)!}$$

Step 2: Total Ways Without Restrictions

If none of the points were collinear, the total number of triangles that could be formed by choosing any $3$ points out of $20$ is:

$$\text{Total combinations}={}^{20}{C}_3$$

Let's calculate this value:

$${}^{20}{C}_3=\frac{20\times 19\times 18}{3\times 2\times 1}$$

$${}^{20}{C}_3=20\times 19\times 3=1140$$

Step 3: Subtract Collinear Combinations

We are told that $5$ points lie on the same straight line (collinear). Any $3$ points chosen strictly from these $5$ collinear points will just form a line segment, not a triangle.

The number of invalid triangles formed by these $5$ points is:

$$\text{Collinear combinations}={}^5{C}_3$$

Let's calculate this value:

$${}^5{C}_3={}^5{C}_2=\frac{5\times 4}{2\times 1}=10$$

Step 4: Calculate Final Number of Triangles

To find the actual number of triangles formed, subtract the invalid collinear cases from the total possible combinations:

$$\text{Number of Triangles}={}^{20}{C}_3-{}^5{C}_3$$

$$\text{Number of Triangles}=1140-10=1130$$

Correct Answer:

(b) $1130$