There are $15$ students. Assume that there are $365$ days in a year.
What is the probability that $3, 5$ and $7$ of them have the same birthday on any three days of a year, with these three groups of students have a different birthday?
$$ P\left(3\right)=\frac{365}{365}\times\frac{1}{365}\times\frac{1}{365}=\frac{1}{133225} $$ $$ P\left(5\right)=\frac{364}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}=5.6187\times{10}^{-11} $$ $$ P\left(7\right)=\frac{363}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}\times\frac{1}{365}=4.206\times{10}^{-16} $$
should I change the denominator for $P(5)$ to $364$ and $P(7)$ to $363$ as after each group the available days for birthday decreases?
Can someone please explain it to me?