The following approximation may be useful.

If there are $k$ people and $N$ possible birthdays (or in case of a hash table, $k$ items being hashed into $N$ buckets), then the expected number of people/items that collide with at least one of the others is exactly (see Henry's answer or André Nicolas's answer)
$$ 
\begin{align} 
& k \left(1 - \left(1-\frac1N\right)^{k-1}\right) \\
& = \frac{k(k-1)}{N} - \frac{k(k-1)(k-2)}{2N^2} + O\left(\frac1{N^3}\right) \\
& \approx \frac{k^2}{N}.
\end{align}$$

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The above is one possible definition of "expected number of collisions". If there are $r$ birthdays/buckets each with two people/items in them, the above expression gives count $2r$, as it counts each member of each pair. If instead you want to count the number of buckets/birthdays that have multiple people in them, then the answer is approximately
$$ \approx \frac{k^2}{2N}.$$

This result can be derived either

* from the previous analysis, by noting that to the first order the most common type of collision is to have 2 in a bucket (3-way and higher collisions will be statistically rare), so you just halve the count;

* or, by doing a similar analysis focusing on birthdays/buckets: the probability that either $0$ or $1$ of the $k$ people have that particular birthday is
$$ \left(1 - \frac1N\right)^k + k\frac1N\left(1 - \frac1N\right)^{k-1}$$
So the expected number of buckets with multiple values in them is
$$
\begin{align}
& N \left(1 - \left(1 - \frac1N\right)^k - k\frac1N\left(1 - \frac1N\right)^{k-1}\right) \\
& = \frac{k(k-1)}{2N} - \frac{k(k-1)(k-2)}{3N^2} + O\left(\frac1{N^3}\right) \\
& \approx \frac{k^2}{2N}.
\end{align}$$