Number Of Triangles From Points

Question

Given 10 points in a plane of which no three are collinear, the number of distinct triangles that can be formed using these points as vertices equals which value?

RankGuru · Accepted Answer

A triangle is determined by choosing 3 non-collinear points, so when no three points are collinear the count is simply a combination, a clean JEE Advanced geometry-counting result. With 10 points and no three collinear, every selection of 3 points forms a valid triangle, and the number of such selections is C(10, 3) = (10 × 9 × 8)/(3 × 2 × 1) = 120. The non-collinearity condition ensures no chosen triple is degenerate. Option 720 = P(10,3) counts ordered triples, overcounting each triangle 6 times. Option 30 miscomputes the combination. Option 100 has no combinatorial basis. Hence there are 120 triangles. Plausibility check: dividing the ordered triple count 10 × 9 × 8 = 720 by 3! = 6 to remove vertex orderings gives 120, matching C(10,3) and confirming that each unordered triple yields exactly one triangle. Distributing distinct objects into distinct boxes assigns each object an independent choice, so the count is an exponential and coincides exactly with the number of functions between the corresponding sets. Recognizing this equivalence between distributions and functions unifies counting problems with the function-counting results from sets and relations, a connection examiners value across chapter boundaries.

Number Of Triangles From Points

Select the correct option:

🔒 Solution Hidden from View

More permutations and combinations Practice Questions

The number of ways to arrange 4 boys and 4 girls in a row so that no two girls are adjacent to each ...

The number of ways to place 5 distinct letters into 3 distinct mailboxes, where any mailbox may rece...

If the combinatorial equation C(n, 2) equals 28 holds for a positive integer n, then the value of n ...

Using the standard combinatorial identity, the value of the sum C(8,0) + C(8,1) + C(8,2) + ... + C(8...

From 20 points placed on a straight line, the number of line segments that can be formed by joining ...

Using all five digits 1, 2, 3, 4 and 5 exactly once to form five-digit numbers, the sum of all such ...

If all the distinct arrangements of the letters of the word RANK are listed in dictionary order, the...

At a party every pair of guests shakes hands exactly once, and the total number of handshakes record...

The number of ways to arrange 5 boys and 3 girls in a row so that all 3 girls always stand together ...

A student must travel from city A to city C passing through city B, with 4 roads from A to B and 5 r...

The number of ways to arrange the letters of the word MONDAY so that all the vowels occupy only the ...

The number of distinct ways in which 8 different people can be seated around a circular table, where...

About This Question

Solution