Question

In a plane, there are 11 points, out of which 5 are collinear. Find the number of straight lines made by these points.
1. 44
2. 47
3. 46
4. 45

Answer 1

Correct Answer - Option 3 : 46

Concept:

If there are n points in plane put of which m (< n) are collinear, then

The total number of different straight lines obtained by joining these n points is

 ⁿC2 – ^mC2 +1

Calculation:

Given that,

n = 11 and m = 5 

⇒ Number of straight lines = ⁿC2 – ^mC2 +1

= ¹¹C₂ - ⁵C₂ + 1

= \(\frac{11!}{2\times (11-2)!}-\frac{5!}{2!\times(5-2)!}+1\)

= \(\frac{11\times 10\times 9!}{2 \times 1 \times 9!}-\frac{5\times 4\times 3!}{2 \times 1 \times 3!}+1\)

= 55 - 10 + 1

= 46

Hence, the number of straight lines will be 46.

1. The total number of different triangles formed by joining these n points is ⁿC3 – ^mC3 
2. The number of diagonals in the polygon of n sides is ⁿC2 – n
3. If m parallel lines in a plane are intersected by a family of other n parallel lines. Then the total number of parallelograms so formed is ^mC2 × ⁿC2
4. The number of triangles formed by joining vertices of the convex polygon of n sides is ⁿC3 
5. Number of triangles having exactly 2 sides common to the polygon = n
6. Number of triangles having exactly 1 side common to the polygon = n(n-4)