Permutations and Combinations

Question

Six distinct points are marked on each of two parallel lines.

 

(i)Calculate the number of distinct quadrilateral which may be formed using 4 of the 12 points are vertices.

 

(ii) distinct triangles which may be formed using 3 of the 12 points as vertices.

Answer 1

Two lines containing points numbered A1 A2 A3 A4 A5 A6

and another line containing points numbered B1 B2 B3 B4 B5 B6.

 

 

To make a quadrilateral, choose any two "A" points--can be done in ₆C₂ ways (this is 15 possibilities).

Let's call the segment joining the two points chosen the "top of the quadrilateral."

 

Now choose any two "B" points--can also be done in 15 ways; from those we draw the segment called "bottom of the quadrilateral."

 

Example  A3   A6

 

            B2   B4.  

 

Just connect the left end point A3 to left endpoint below B2, and similarly connect the two right endpoints. In other words, just finish drawing the quadrilateral.

 

15 times 15 is 225 possible quadrilaterals.

 

I leave to you the task of figuring out the answer to the triangles question, #2.