(Q). Consider a square with 5 points on each side (no point on vertices)
(i). How many straight lines can be drawn from these 20 points such that each line passes through exactly 2 of the given points?
(ii). How many triangle can be drawn from these 20 points as vertices?
(iii). How many quadrilateral can be drawn from these 20 points as vertices?
Solution:-
(i).
From 4 sides we have to select 2 sides that can be done in ⁴C₂ ways.
Now from each of these selected sides we have to select 1 point out of 5 points and that can be done in ⁵C₁×⁵C₁ ways.
So total number of such straight lines
= ⁴C₂×⁵C₁×⁵C₁ Answer
(ii).
From 20 points without any restriction we can get maximum ²⁰C₃ triangles.
From this we have to reduce the number of triangles that we can not get since 4 sides has 5 collinear points each, which is equal to 4.(⁵C₃)
So total number of required triangle is
= ²⁰C₃ – 4.(⁵C₃) Answer
(iii).
From 20 points without any restriction we can get maximum ²⁰C₄ quadrilaterals.
From this we have to reduce the number of quadrilateral that we can not get if three points are collinear & this can happen in two cases:
case(i): when all the four selected points are collinear
⁴C₁×⁵C₄ = 4×5 = 20
case(ii): when three points are collinear
⁴C₁×⁵C₃׳C₁×⁵C₁ = 4×10×3×5 = 600
So total number of such quadrilateral = ²⁰C₄ – 20 – 600
= ²⁰C₄ – 620 Answer