Before we start with calculations we should think about which values of $r$ we should take into account.
$r=1,2$: We can exclude the trivial cases $r=1$ and $r=2$ which give a probability of zero, since we never reach a configuration with three people having the same birthday.
$2<r\leq 365$: The main part where we put the focus of our analysis. Fortunately the reasoning can be easily extended to $r\leq 730$.
$365<r\leq 2\cdot 365$: Assuming the year has $365$ days, the situation becomes slightly different if the number of people is greater than $365$. In these cases it is guaranteed that there are people with the same birthday and this needs some additional thoughts.
$r>2\cdot 365$: According to the pigeonhole principle groups with more than $730$ people have at least three equal birthdays with probability $p=1$.
Starter:
In order to get a first impression and to have a verification for small numbers at hand we look at a smaller example. It should be small enough to make calculations easy and large enough to be representative for the problem.
We take $r=5$ people and instead of $365$ days we use $6$ days. If we take a fair die with six faces instead, this smaller version of the problem can then be stated as:
Small Problem: What is the probability that in a group of $5$ people, each of them throwing a fair die once, at least three people roll the same number.
Five people can throw a total of $6^5=7776$ different configuration of $5$-tuples from $(1,1,1,1,1)$ to $(6,6,6,6,6)$.
We answer the question by counting the $5$-tuples which have no more than two equal faces.
- Let $N_{(11111)}$ be the number of $5$-tuples with pairwise different entries. We see that there are $6\cdot5\cdot\ldots\cdot1=6!$ possibilities and get
\begin{align*}
N_{(11111)}=6!=720
\end{align*}
The index $11111$ indicates that there are five pairwise different values counted.
Now we consider the number of $5$-tuples with one or more pairs of equal numbers, but with no occurrence of three or more equal numbers. There are two different constellations:
One pair and three other numbers which are pairwise different, denoted by $N_{(1112)}$.
The second constellation are two different pairs and another number different to each of them, denoted by $N_{(122)}$.
We obtain
\begin{align*}
N_{(1112)}&=\frac{1}{1!}\binom{5}{2}6\cdot5\cdot4\cdot3=3600\tag{1}\\
N_{(122)}&=\frac{1}{2!}\binom{5}{2}6\binom{3}{2}5\cdot4=1800\tag{2}
\end{align*}
Comment:
In (1) there are $\binom{5}{2}$ pairs which can take the values $1,\ldots,6$. The other three people have $5,4$ and $3$ possibilities to throw a number pairwise different to all the others.
In (2) there are $\binom{5}{2}\cdot6$ possibilities for one pair, leaving $\binom{5-2}{2}\cdot5$ possibilities for the second pair. Since we don't want to count different pair constellations more than once, we have to divide them by $2!$.
The distribution table of all different $5$-tuples can be easily calculated
\begin{array}{cccccccc}
\text{Total}&N_{(11111)}&N_{(1112)}&N_{(113)}&N_{(122)}&N_{(14)}&N_{(23)}&N_{(5)}\\
7776&\color{blue}{720}&\color{blue}{3600}& 1200& \color{blue}{1800} & 150 & 300 & 6
\end{array}
We are ready to calculate the wanted probability $p$ for this small example:
\begin{align*}
p&=1-\frac{N_{(11111)}}{6^5}-\left(\frac{N_{(1112)}}{6^5}+\frac{N_{(122)}}{6^5}\right)\\
&=1-\frac{720}{6^5}-\frac{5400}{6^5}\\
&=1-\frac{6120}{7776}\\
&=0.212963
\end{align*}
The real problem:
Most of the work has been done by the small example. If we now consider $365$ days and $r$ people we can go on straight forward.
At first we put the focus at
\begin{align*}
3\leq r \leq 365
\end{align*}
- Let's denote with $N_{(1^{r})}$ the number of $r$-tuples with all entries pairwise different. The exponent in the index $1^{r}$ denotes the multiplicity of $1$.
We obtain
\begin{align*}
N_{(1^{r})}=\frac{365!}{(365-r)!}\tag{3}
\end{align*}
Next we have to consider $r$-tuples containing $k$ pairs of tuples, with $1\leq k\leq \lfloor\frac{r}{2}\rfloor$ together with $r-2k$ single numbers pairwise different to all other single numbers and numbers within tuples. We denote them with $N_{(1^{r-2k}2^k)}$.
We obtain analogously to (2)
\begin{align*}
N_{(1^{r-2k}2^k)}&=\frac{1}{k!}\binom{r}{2}365\binom{r-2}{2}364\cdots\binom{r-2(k-1)}{2}\left(365-(k-1)\right)\\
&\qquad\cdot(365-k)\cdots((365-k)-(r-2k-1))\\
&=\frac{1}{k!}\binom{r}{2}\binom{r-2}{2}\cdots\binom{r-2(k-1)}{2}\frac{365!}{(365-(r-k))!}\\
&=\frac{1}{k!}\frac{r!}{2^k(r-2k)!}\frac{365!}{(365-(r-k))!}\tag{4}
\end{align*}
Plausibility check: If we compare the expression (4) with the small example above we could do a simple check. If we set
\begin{align*}
r=5,k=2,\text{ and }365\rightarrow 6
\end{align*}
we obtain
\begin{align*}
N_{(1^{5-4}2^2)}=N_{(12^2)}=\frac{1}{2!}\frac{5!}{2^2(5-4)!}\frac{6!}{(6-(5-2))!}
=1800
\end{align*}
in accordance with the table entry $N_{(122)}$ above.
The range $365<r\leq 2\cdot 365$
If $r>365$ it is guaranteed that at least two people have the same birthday. It follows
$$N_{(1^r)}=0$$
We see after short consideration, the approach (4) for a group of people with $1\leq k\leq \lfloor r\rfloor$ distinct pairs is also valid up to $r\leq 730$.
Now it's time to harvest. In order to find the number of all $r$-tuples we have to exclude, we sum up $N_{(1^r)}$ and $N_{(1^{r-2k}2^k)}$ for $1\leq k \leq \lfloor\frac{r}{2}\rfloor$.
In fact, inspired by the comment of @robjohn, we can also respect all other values of $r$ in the same formula, if we take $\frac{1}{n!}=0$ for $n<0$ and finally obtain
The probability $p$ that at least three people from a group of $r\geq 1$ people have the same birthday is according to (3) and (4)
\begin{align*}
p&=1-\frac{1}{365^r}\left(N_{(1^r)}+\sum_{k=1}^{\lfloor\frac{r}{2}\rfloor}N_{(1^{r-2k}2^k)}\right)\\
&=1-\frac{365!}{365^r}\left(\frac{1}{(365-r)!}
+r!\sum_{k=1}^{\lfloor\frac{r}{2}\rfloor}\frac{1}{2^kk!(r-2k)!(365-(r-k))!}\right)\\
&=1-r!\frac{365!}{365^r}\sum_{k=0}^{\lfloor\frac{r}{2}\rfloor}\frac{1}{2^kk!(r-2k)!(365-(r-k))!}
\end{align*}
