Question

Any 2 points determine a line. If there are 6 points in a plane, no 3 of which lie on the same line, how many lines are determined by pairs of these 6 points?

The answer key says 15, and the explanation is (6!)/(2!4!). Could someone explain to me why there is a 6, 2, and 4 factorials and how to approach this question please?

Answer 1

Paste your question in google.

You will find couple explanations.

Answer 2

This type of question often comes up when you study
"permutations" and "combinations"
You clearly must be studying permutations and combinations

In this case you are "choosing" any 2 of the 6 points
choose ----> combinations, (the order in which you choose any 2 points does not matter)
e.g. AB or BA yields the same line

the definition of C(n,r) is n!/(r!(n-r)!) , see your text

and we have to choose 2 from 6
= C(6,2) = 6!/(2!4!) = 15