There are $20$ children in a lost ship. They do not remember their birthdays but would like to be assigned with one.
1. In how many ways this can be done so that exactly $2$ children will get identical day and the rest will get each a separate day?
2. What is the number of possibilities to assign, so that at least one day of the year will belong to at least $2$ children?
My attempt
For the first one, I reckon that we could choose $2$ out of $20$ children so that is $^{20}\mathrm{C}_2$. And for the birthdays, we would have $19$ entities as $2$ people would share the same birthday. That would result to $^{365}P_{19}$. And we multiply those together to get the answer. Is that correct?
For the second one, is it correct if $365^{20} -\; ^{365}P_{20}$? I am subtracting all the possibilities with the complement of at least one day of the year, at least $2$ children will have the same birthday which means that no children at all share the same birthday.