A man invited five friends.

Expanding on the comments...

There are a total of $30$ days, where each day is independent of the other. The probability that the host is born on the $n_{th}$ day is $Pr(X = n) = \frac{1}{30}$, if the host is equally likely to be born on any day of this month. The probability that each friend is born on the $n_{th}$ day is also $\frac{1}{30}$ and similarly not on the $n_{th}$ day, is $\frac{29}{30}$. So it would be $(\frac{29}{30})^5$. As for $1 - (\frac{1}{30})^5$, this just means your including everything else except the all of the his friends being born on the same day. Note that set does not just include the events: all the friends being born on the same day or none of them being born on the same day.

Alternatively, the set is binomially distributed. What if $4$ friends are born on the same day and $1$ is not or $2$ friends are born on the same day and $3$ are not?

In mathematical notation your stating that, $$Pr(X = \text{None of his friends}) = 1 - Pr(X = \text{All of his friends})$$ $$Pr(X = 0) = 1 - Pr(X = 5)$$ This is incorrect as $$Pr(X = 5) = 1 - Pr(X \leq 4), \text{and }$$ $$Pr(X = 0) = 1 - Pr(X \ge 1), \text{and }X\sim Bi(n, p)$$