Pigeonhole principle - how many people to get 2 with the same initials

Question

I am trying to solve the following problem:

There are 800 seats in the cinema. How many seats needs to be occupied in order to have at least 2 people with the same initials (first name and last name) in the cinema?

I think it's a problem for the pigeonhole principle. I know that there are 26 characters in the alphabet. How can I use the pigeonhole principle? I don't know how to start.

Thanks

Answer 1

There are $26$ letters in the English alphabet. That means there are $26$ choices for the first initial, and $26$ choices for the second initial. As such there are $$ 26\cdot 26=676 $$

possible initials a person could have. That means that, via the pigeonhole principle, we could only have $676$ people in a room with unique initials. Once we go over that number, we are guaranteed that two people must share initials