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Expert: - 7/6/2009


Question
Hi,

I don't understand this.... Find a relationship between entries in Pascal's triangle and the maximum number of diagonals in a n-sided polygon.

Answer
Now Pascal's triangle I'll start with row 0, which is 1.
Row 1 can be seen to be 1 1.

Row 2 can be seen to be 1 2 1.

Row 3 can be seen to be 1 3 3 1.
Look at the 3rd element in that row.  It is 3.
Look at a triangle.  There are 3 points to join the corners.

Row 4 can be seen to be 1 4 6 4 1.
Look at the 3rd element in that row.  It is 6.
Look at a 4 sided shape (a square).  
There are 4 sides and 2 diagonals for a total of 6 lines.
That is the 3rd number in that row.

Row 5 can be seen to be 1 5 10 10 5 1.
Look at the 3rd element in that row.  It is 10.
For a pentagon, there are 5 sides and a star in the middle for five lines in the middle.  That's 5 + 5 lines, which is 10 lines.
That is the same as the 3rd element in the 5th row of Pascal's triangle.

Consider of shape with n corners.  There are n-1 lines from each corner and n corners, but note that each line as 2 ends, so take
n(n-1)/2 to get the number of connecting line.  If the nth row of Pascal's triangle is looked at, that can be seen to be the third element in that row.  The first one is 1, the second one is n, and the third one is n(n-1)/2.

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