Geometric Applications of PNC

Q. On a triangle ABC, on the side AB, 5 points are marked, 6 points are marked on the side BC and 3 points are marked on the side AC (none of the points being the vertex of the triangle). How many triangles can be made by using these points?

1. 90
2. 333
3. None of these
4. 328

Q. Two straight lines intersect at a point O. Points $A_1,A_2,A_3,A_4,A_5,...,A_m$ are taken on one line and points $B_1,B_2,B_3,....,B_n$ on the other. If the point O is not included, the number of triangles that can be drawn using these points as vertices, is:

1. None of these
2. ${}^n{C}_2{+}^m{C}_2$
   
3. ${}^{2n}{C}_2$
   
4. ${}^{m+n}{C}_2$

Q. Two lines intersect at O. Points $A_1,A_2,....A_n$ are taken on one of them and $B_1,b_2,...B_n$ on the other the number of triangles that can be drawn with the help of these (2n+1)ponints is:

1. n
2. $n^2$
3. $n^3$
4. $n^4$

Q. Three are n points in a plane, no three being collinear except m of them which are collinear. The number of triangles that can be drawn with their vertices at three of the given points is

1. ${}^n{C}_3{-}^m{C}_3$
2. None of these
3. ${}^{n-m}{C}_3$
4. ${}^n{C}_3-m$

Q. The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is

1. 24
2. 52
3. 48
4. 16

Q. Eight straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. The number of parts into which these lines divide the plane is:

1. 17
2. 37
3. 73
4. 72

Q. Maximum number of points of intersection of 6 circles, is:

1. 30
2. 28
3. 15
4. None of the these

Q. The straight line $I_1,I_2,I_3$ ar parallel and lie in the same plane. A total number of m points on $I_1$, n points on $I_2$, k points on $I_3$ are used to produce the triangles, the maximum number of triangles formed with vertices at these points are:

1. None of the above
2. ${}^m{C}_3{\times }^n{C}_3{\times }^k{C}_3$
   
3. ${}^{(m+n+k)}{C}_3-{(}^m{C}_3{+}^n{C}_3{+}^k{C}_3)$
   
4. ${}^{(m+n+k)}{C}_3$
   

Q. Maximum number of points into which 3 circles and 3 lines intersect is:

1. 21
2. 9
3. 27
4. 3!

Q. Two lines intersect at O. Points $A_1,A_2,....A_n$ are taken on one of them and $B_1,B_2,...B_n$ on the other. The number of triangles that can be drawn with the help of these (2n+1) points is:

1. n
2. $n^4$
3. $n^2$
4. $n^3$