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Question: if 9 points are given with 5 points in a line, how many lines can be formed between two points?...

if 9 points are given with 5 points in a line, how many lines can be formed between two points?

A

27

B

36

C

26

D

30

Answer

27

Explanation

Solution

Let N be the total number of points and K be the number of collinear points.
Given N = 9 and K = 5.

The total number of lines that can be formed by joining any two points out of N points, assuming no three points are collinear, is given by the combination formula NC2^N C_2. 9C2=9!2!(9−2)!=9×82×1=36^9 C_2 = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36.

However, K points are collinear. These K collinear points form only one distinct line. If we choose any two points from these K collinear points, they will lie on the same line. The number of pairs of points from these K collinear points is KC2^K C_2. 5C2=5!2!(5−2)!=5×42×1=10^5 C_2 = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10.

The initial calculation 9C2^9 C_2 includes the 5C2^5 C_2 pairs of points chosen from the collinear points. These 10 pairs all form the same single line. So, in the count of 36, this single line has been counted 10 times (once for each pair). To get the number of distinct lines, we subtract the overcounted lines (5C2^5 C_2) and add 1 for the single line formed by the collinear points.

Number of distinct lines = (Total lines assuming no collinearity) - (Lines formed by collinear points) + (Single line formed by collinear points)
Number of distinct lines = NC2−KC2+1^N C_2 - ^K C_2 + 1
Number of distinct lines = 9C2−5C2+1^9 C_2 - ^5 C_2 + 1
Number of distinct lines = 36−10+136 - 10 + 1
Number of distinct lines = 26+1=2726 + 1 = 27.

Alternatively, we can consider the types of lines formed:

  1. Lines formed by choosing 2 points from the 5 collinear points: These form 1 distinct line.
  2. Lines formed by choosing 1 point from the 5 collinear points and 1 point from the remaining 9−5=49-5=4 non-collinear points: 5C1×4C1=5×4=20^5 C_1 \times ^4 C_1 = 5 \times 4 = 20 distinct lines. Each such line connects a point on the collinear line to a point not on it, so each line is distinct from the collinear line and from other lines of this type (unless some of the 4 points are collinear among themselves or with some of the 5 points in a different collinear arrangement, which is not stated).
  3. Lines formed by choosing 2 points from the 4 non-collinear points: 4C2=4×32=6^4 C_2 = \frac{4 \times 3}{2} = 6 distinct lines, assuming no three of these 4 points are collinear.

Total number of distinct lines = 1+20+6=271 + 20 + 6 = 27.

Both methods yield the same result.