If all you care about is the most likely number of matches, you don’t actually have to calculate large exponentials. The probability that there are exactly n matches is hypergeometric, as already stated; we can write it as
P(n) = (A! B! (N-A)! (N-B)!) / (n! N! (A-n)! (B-n)! (N-A-B+n)!)
(here N=60000, A=6100, B=3500). P(n) increases to the most likely number of matches n[sub]0[/sub] and then decreases; so P(n)/P(n-1) is >=1 for n<=n[sub]0[/sub] and <1 for n>n[sub]0[/sub]. But
1 <= P(n)/P(n-1) = ((A-n+1) (B-n+1)) / (n (N-A-B+n))
implies
n <= (A+1)(B+1)/(N+2)=355.981…
so we have n[sub]0[/sub]=355.
Routines for calculating hypergeometric probabilities are found in many software packages. Excel even has such a function, called hypgeomdist.