How many people have to be gathered to ensure at least 9 people have the same first letter of their first name?

Question

I know I use generalized pigeon hole principle where $n/k= m$. I know $n$ is pigeons and $k$ is pigeonholes. I know I have to do n/#=m for this one. So its n/#=9. Im not sure what would be pigeonholes in this case.

Answer 1

34

26 people can have different first letters of their names. The 27th person has to share, then you have 2 people sharing the first letter. If a new person (the 28th) is added with a different letter than the already shared ones, then 2 more people share a letter, but if the new person has the same first letter as the other two who are already sharing then only 1 more person shares a letter. So in order to ensure that 9 share, then 27 (with 2 sharing) plus 7 more is 34 with 9 sharing.

Adjust this if there is a different alphabet with something other than 26 starting letters

Answer 2

If I understand your question, the number is $26\times 8 +1=209$.