There are 5 points P1, P2, P3, P4, P5 on the side AB, excluding A and B, of a triangle ABC. Similarl

Q: Wait, why can't I just pick one point from each side? Wouldn't that always form a triangle?

That is a great observation! Picking one point from each side will indeed form a triangle. However, that is only one specific way to form a triangle. You could also pick two points from one side and one from another. Using the complementary counting method (Total - Invalid) is much more robust and ensures you do not miss any valid combinations.

Q: Why do we subtract the collinear cases? Can't we just add up the valid combinations?

You absolutely could add up all the valid cases, but that would require calculating many different scenarios (e.g., 1 from AB, 1 from BC, 1 from CA; or 2 from AB, 1 from BC, etc.). Subtracting the invalid cases is the standard, elegant JEE approach because it simplifies the calculation significantly.

Q: Does the order of selection matter here?

No, it does not. Choosing points {P1, P2, P3} is exactly the same as choosing {P3, P2, P1}. Since the order of selection does not change the triangle formed, we use combinations (${}^{n}C_{r}$) rather than permutations.

Question

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $AB$, excluding $A$ and $B$, of a triangle $ABC$. Similarly there are 6 points $P_6, P_7, \dots, P_{11}$ on the side $BC$ and 7 points $P_{12}, P_{13}, \dots, P_{18}$ on the side $CA$ of the triangle. The number of triangles, that can be formed using the points $P_1, P_2, \dots, P_{18}$ as vertices, is :

Options:

(A) 776

(B) 796

(C) 751

(D) 771

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To determine the number of unique triangles that can be formed, we first identify the total number of points available on the sides of triangle $ABC$. We are given 5 points on $AB$, 6 points on $BC$, and 7 points on $CA$.

The total number of points is:
$$5 + 6 + 7 = 18$$

### The Total Universe of Possibilities

To form a triangle, we must select any 3 points from the total set of 18. The number of ways to choose 3 points from 18 is given by the combination formula ${}^{18}C_{3}$.

The calculation is as follows:
$${}^{18}C_{3} = \frac{18 	imes 17 	imes 16}{3 	imes 2 	imes 1} = 3 	imes 17 	imes 16 = 816$$

This value, 816, represents our total universe of selections. However, we must account for the fact that not every selection of 3 points results in a valid triangle.

### The Collinearity Trap

In geometry, three points form a triangle only if they are non-collinear. If we select 3 points that lie on the same side of the triangle, they form a straight line rather than a triangle. This is a classic constraint in combinatorial geometry.

We must calculate the number of invalid cases for each side:

For side $AB$ (5 points):
$${}^{5}C_{3} = \frac{5 	imes 4 	imes 3}{3 	imes 2 	imes 1} = 10$$

For side $BC$ (6 points):
$${}^{6}C_{3} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$$

For side $CA$ (7 points):
$${}^{7}C_{3} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = 35$$

### The Final Synthesis

To find the number of valid triangles, we subtract the total number of collinear (invalid) cases from the total universe of selections. The sum of invalid selections is:
$$10 + 20 + 35 = 65$$

Subtracting these from the total:
$$816 - 65 = 751$$

By applying the principle of complementary counting, we conclude that the number of unique triangles that can be formed is **751**.

Select Answer:

Visualized Solution

Visualizing the Triangle

Placing the Points

Total Available Points

Total Possible Selections

The Collinearity Trap

Subtracting Invalid Cases

Calculating Total Combinations

Calculating Collinear Cases

Final Calculation

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Total Universe of Possibilities

The Collinearity Trap

The Final Synthesis

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Select Answer:

Visualized Solution

Visualizing the Triangle

Placing the Points

Total Available Points

Total Possible Selections

The Collinearity Trap

Subtracting Invalid Cases

Calculating Total Combinations

Calculating Collinear Cases

Final Calculation

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Total Universe of Possibilities

The Collinearity Trap

The Final Synthesis

Similar Questions

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