the formula I worked for the probability that no two or more people will share the same birthday. (no leap years)
$$\frac{365!}{365^n (365-n)!}$$
But what is least number of people such that the probability is at least 0.5? whats the way to calculate this without testing 1 by one?
This is a well known problem called the BIRTHDAY PROBLEM.
You can find all the maths here
https://en.wikipedia.org/wiki/Birthday_problem
If you know the probability for $n$ people to have the same birthday, you can immediately fine the probability for $n$ people to have different birthday.