ProblemWhat is the chance that at least two people in a group of k people will have the same birthday?

By same birthday we mean same day and month but not necessarily
the same year. We assume further that the birthdays of the k people
are unrelated and each equally likely to be in any of the 365 days
of the year.
SolutionDefine the proposition:E = "at least 2 equal birthdays"Pr(ES) = 1  Pr(not E S)(not E) = "all different birthdays"365*364*...*(365k+1).365^{k} 
> answer := 1  a!/((ak)!*a^k);
365! answer := 1   k (365  k)! 365
This is a function of k that can be easily computed with 
> p := k > evalf((1  365!/((365k)!*365^k)),4):
> p10 := p(10); p20:=p(20); p23 := p(23); p40:=p(40); p50:=p(50);
p10 := .1169 p20 := .4114 p23 := .5073 p40 := .8912 p50 := .9704