Pascal's Triangle: A number of long, thin sticks are lying

[lisa.]

A number of long, thin sticks are lying in a pile at odd angles such that the sticks cross each other.

a) Relate the maximum number of intersection points of n sticks to entries in Pascal's triangle.

b) What is the maximum number of intersection points with six overlapping sticks?

thanks for the help

 

[arthur ohlsten]

b)
1.place one stick on xy plane in a on y axis
2. place second stick on plane so it differs by 1 degree. It crosses at one point
3. Place third stick at two degree angle relative to first, thus it crosses both sticks
4. place 4th stick at 3 degrees so that it crosses three sticks
5. place 5th stick at 4 degrees so it crosses the other 4 sticks
6. place 6th stick at 5 degrees so it crosses othe 5 sticks

number of intersections= 1*2*3*4*5
no.=5!

a)
number of intersections of n sticks is [n-1]!
Coefficients of Pascals triangle are kCn=n!/[k![n-k]!]
for k=1 and n=[n-1]
[n-1]!/[n-2]!

or the number of intersections =[n-2]! times the Pascal term for [n-1] things taken 1 at a time

I hope this is right
Arthur