you're close on the math.
(ignoring leap-years) the probability that n people all have different birthdays is: ((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365) = 365! / ((365-n)! * 365^n)
setting that equal to 50% and solving is probably harder than just plugging in numbers on a calculator till you get it. 23 is the magic number.