Pigeonhole again

Question

There are $10$ bird cages with the maximum of $5$ birds inside it. How many birds should be prepared so that I can be sure that there are $3$ cages with $2$ birds inside it?

My answer :

I fitted all the cages with $1$ bird, that means $10$ and I filled the other $3$ cages with $1$ more bird. That makes it $13$ And everyone said it isn't $13$, to be sure I need $19$ birds. Can someone make me understand?

Thanks

Answer 1

You can fill all 10 cages with 1 bird (as you did), but then fill two cages to capacity (so another 8 birds) without having 3 cages with 2 birds.