Tricky probability question which cant be solved using exclusion?

Question

I am confused on how to go about solving this problem-

" What is the probability that 2 people in the group have a birthday in the same month out of a)exactly 20 people? b)atleast 20 people"

I think answer to b) part is 1 using pigeon hole principle-correct me if i am wrong.

For a) part According to me principle of inclusion and exclusion will work till n=12

BUT what to do after that?

Answer 1

I'm not sure if this is answering your question but I will just place it here anyway.

Let the set $A$ = The group of people. Such that $|A|= 20$

Let the set $B$ = Number of months. Such that $|B|=12$

The generalized pigeon-hole principle in terms of sets states that if $f:A\rightarrow B$ from finite set $A$ to finite set $B$, with $|A|= n$ and $|B|= m$; such that $n \gt m$ then there exists at least one element in $B$ such that $f$ maps $\left\lceil\frac{n}{m}\right\rceil$ elements of $A$ to it.

In your case $n =20$ and $m =12$ so $\exists$ at least $\left\lceil\frac{20}{12}\right\rceil=2$ people that have their birthday in the same month.