A Counting Problem

(x^4-6x^3+23x^2-18x+24)/24

I think that is the correct answer, though you should realize that you can express it in a more elegant form (namely as the sum of binomial coefficients. What is your reasoning/how did you get the answer? Did you assume it was a polynomial and interpolated?

C(n, 4) + C(n, 2) + 1

Number of lines drawn between n points can be expressed as a series (n-1)+(n-2)+...(n-(n-1))

Or more practical as binomial coefficient: C(n,2). Expressing number of possibilities to connect 2 points in an n point circle by a line.

C(n,4) gives the number of possible intersects between any 4 of the n points in the circle.

Adding them together and the extra region in the center (+1), gives you the total.

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That is the correct answer Can you justify why your answer is correct though? As in why should counting these intersections give you the number of regions?

Every time a line intersects another line, you bisect two regions into four regions