Probability of an exact number of duplicate pairs when choosing X from Y. Here is the problem I'm faced with, as best as I can describe it.
There is a set of 256 values (a byte), and 108 values are chosen from this set. Each choice may be any value from 0 to 255. What is the probability that once the values are chosen, there will be six distinct pairs of duplicate values, and all other 96 values will be unique?

You have $6+96+154=256$ and $6\times 2 + 96 \times 1 + 154 \times 0 = 108$ so the probability you are looking for is 
$$\frac{\frac{256!}{6!\;96!\;154!} \times \frac{108!}{(2!)^6\;(1!)^{96}\;(0!)^{154}}}{256^{108}} $$
Some more background on this approach can be found at https://math.stackexchange.com/a/23282/6460 and https://math.stackexchange.com/a/23696/6460 
The numerical calculation may stretch some calculators.  I make it about  $6.54 \times 10^{-6}$. 
