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?

First let us compute the number of choices in a canonical order.  We list the pairs first in increasing order, then the single numbers.  There are $\binom {256}{6}$ ways to pick the pairs and $\binom {250}{96}$ ways to pick the singles.  Given a set of numbers, they can be ordered in $\frac {108!}{2^6}$ ways, as interchanging any of the pairs is equivalent.  The total probability is then $\frac {\binom {256}{6}\binom {250}{96}108!}{256^{108}2^6}\approx 6.5E-6$
