Question

Given 12 points in a plane, no three of which are collinear. then number of line segments can be determined, are:

a. 20
b. 54
c. 120
d. 66

Answer 1

The number of line segments that can be formed with 12 points in a plane where no three points are collinear is 66, determined by using the combination formula C(12, 2). The correct answer is d. 66.

Step-by-step explanation:

To determine the number of line segments that can be formed using 12 points in a plane where no three points are collinear, you can use the combination formula for selecting 2 points at a time from the 12 points. The combination formula is expressed as C(n, k) = n! / (k! * (n-k)!), where n is the total number of points and k is the number of points to form a line segment (which is 2 in this case).

1. First, substitute the values into the combination formula: C(12, 2) = 12! / (2! * (12-2)!).
2. Simplify the factorial expressions: 12! = 12 x 11 x 10!, and (10!) cancels out in both the numerator and the denominator, leaving you with 12 x 11 / 2!.
3. Calculate the result: 12 x 11 / (2 x 1) = 132 / 2 = 66.

Therefore, the number of line segments that can be formed is 66, which corresponds to option (d).