BrowserUk:

In the iterative solution, we're accumulating f(x)=(1-e^(-x/N))^h over the range of x=0 .. NumSamples. That's a rough form of computing the definite integral of the expression. Integrating over three variables (x, N, h) would be a pain, so I treated N and h as constants.

So first, we multiply out our f(x) expression to remove the exponent. So using h as 2, we get:

f(x) = 1 - 2e^(-x/N) + e^(-2x/N)

integ(f(x)) = integ( 1 - 2e^(-x/N) + e^(-2x/N) )
            = integ( 1 ) - 2*(integ(e^(-x/N))) + integ(e^(-2x/N))

since:
integ(1) = x + C
integ(e^(bx)) = (1/b)e^(bx) + C

integ(f(x)) = [ x+C ] + [ (-2N)e^(-x/N) + C ] + [ (-N/2)e^(-2x/N) + C 
+]
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Computing a definite integral over a range is simply integ(f(x)) evaluated at the upper limit less the value evaluated at the lower limit. This causes the C terms to cancel.

Pascal's triangle comes out because we've got (a+b)^n, and when we multiply it out, we get the binomial expansion which is where the coefficients come into play.

One point I should mention: You don't have to use 0 as the lower bound. If you wanted the number of collisions you'd experience from sample A to sample B, just evaluate integ(f(B))-integ(f(A)). By using A=0 we compute the number of collisions for the entire run.

...roboticus

When your only tool is a hammer, all problems look like your thumb.