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

You can ignore the first sentence in the problem (if you didn't figure that out already). — Andrew Chin, Jan 06 '20 at 18:22
Ok, so we could fill $676$ seats without ever duplicating initials. What about $677$? — lulu, Jan 06 '20 at 18:24

score 1 · Accepted Answer · answered Jan 06 '20 at 18:26

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

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

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