Let's say that I have a set of unique elements, $P$, and a multiset $M$ that I fill with $N \leq ||P||$ elements by sampling with replacement from $P$. What is the probability that the multiset $M$ contains $0 \leq k \leq N$ elements that occur once (i.e. elements with only a single copy in the multiset)?
The probability of a single element in a multiset of cardinality $N$ occurring only once should be equivalent to tossing $N$ balls into $||P||$ bins, and finding the probability that a particular ball is by itself in a bin.
To get started...
From pg. 95 of "Probability and computing: Randomized algorithms and probabilistic analysis" by Michael Mitzenmacher and Eli Upfal, when we toss $N$ balls into $||P||$ bins, the probability that a specific bin has exactly $r$ balls, P[$r$], is given as:
P[$r$] = ${N \choose r}$ $(\frac{1}{||P||})^r(1-\frac{1}{||P||})^{(N-r)}$
By linearity of expectation, we can now write an expression for the expected number of balls that exist in a bin of $r$ balls as: E[X] = $||P||*r*{N \choose r}$ $(\frac{1}{||P||})^r(1-\frac{1}{||P||})^{(N-r)}$
For $r = 1$, $E[X] = N*(1-\frac{1}{||P||})^{N-1}$