Suppose that there are $5$ red points and blue points on a circle.Find the number of convex polygons whose vertices are among the $9$ points and having at least one blue vertex.

Open in App

Solution

The total no. of polygons =
$(\frac{9}{3}) + (\frac{9}{4}) + . . . + (\frac{9}{9}) = 2^{9} - 1 - 9 - 36 = 466$
Number of polygons with only red vertices=
$(\frac{5}{3}) + (\frac{5}{4}) + (\frac{5}{5}) = 16$
The number of convex polygon satisfying the given condition $= 466 - 16 = 450$

Suggest Corrections

Similar questions

Suppose $2016$ points of the circumference of a circle points are coloured red and the remaining points are coloured blue. Find the minimum possible value of a natural number $n$ , for which there exists a regular $n$ - sided polygon whose all vertices are blue.

Q. Suppose that the vertices of a regular polygon of

20

sides are coloured with three colours {red, blue and green} such that there are exactly three red vertices. Prove that there are three vertices

A, B, C

of the polygon having the same colour such that triangle ABC is isosceles.

Q. There are two parallel lines, one having

10

points and the other having

5

points. The number of triangles formed with vertices as these points is

Q. In an even

n

-sided polygon, the sides are painted blue and red alternatively. The vertices of the polygon are joined to form triangles with one side in common with the polygon. Find the number of such triangles with one blue side.

Q. There are five different green dyes, four different blue dyes and three different red dyes. The total number of combinations of dyes that can be chosen taking at least one green and one blue dye is

Join BYJU'S Learning Program

Suppose that there are 5 red points and blue points on a circle.Find the number of convex polygons whose vertices are among the 9 points and having at least one blue vertex.

The total no. of polygons =(93)+(94)+...+(99)=29−1−9−36=466Number of polygons with only red vertices=(53)+(54)+(55)=16The number of convex polygon satisfying the given condition=466−16=450

Suppose that there are $5$ red points and blue points on a circle.Find the number of convex polygons whose vertices are among the $9$ points and having at least one blue vertex.

The total no. of polygons =
$(\frac{9}{3}) + (\frac{9}{4}) + . . . + (\frac{9}{9}) = 2^{9} - 1 - 9 - 36 = 466$
Number of polygons with only red vertices=
$(\frac{5}{3}) + (\frac{5}{4}) + (\frac{5}{5}) = 16$
The number of convex polygon satisfying the given condition $= 466 - 16 = 450$