Question

Q11 There are 15 points in a plane, of which exactly 8 are collinear. If p is the number of straight lines that can be formed using these points, while q is the number of triangles that can be formed using these points, then the value of q = p is equal to​

Answer 1

Answer:

(i) No. of lines =

12

C

2

-

5

C

2

+ 1 = 66 - 10 + 1 = 57.

(ii) No. of Δ

s

=

12

C

3

-

5

C

3

= 220 - 10 = 210.

Answer 2

Answer:

Number of triangles that can be formed is equal to the number of ways to select 3 non-collinear points.

Number of ways to select 3 points from 15 points =

15

C

3

Let n points be collinear.

Number of ways to select 3 points out of the n collinear points=

n

C

3

So, Number of ways to select 3 non-collinear points = Number of ways to select 3 points using all the points - Number of ways to select 3 points using the collinear points

So, Number of ways to select 3 non-collinear points =

15

C

3

−

n

C

3

So, Number of triangles that can be formed =

15

C

3

−

n

C

3

∴445=

15

C

3

−

n

C

3

∴445=455−

n

C

3

∴

n

C

3

=10

∴

3!(n−3)!

n!

=10

∴

6

n(n−1)(n−2)

=10

∴n(n−1)(n−2)=60

Solving this equation we get n=5

Hope it helps

✌️Santa19 ✌️