This isn't right on three counts:
1. This calculation pretends that a person who is a descendant of one pair of nth-great grandparents will NOT be also a descendant of other pairs of nth-great grandparents. This is clearly not the case.
Take, for example, the case of 3rd cousins, and where the number of children = 2. The above formula solves to 128 third cousins. But that's not right, because the total number has to be reduced for all the examples of multiple counting. For the 8 breeding pairs of ancestors that would lead to 3rd cousins, the subject would appear in the descendants list of all 8 pairs, not just one. So that total number needs to be reduced by 8-1=7. Similarly the total needs to be reduced by 7 to account for the sibling of the subject. Then there would be 4 first cousins who would appear in 4 of the 8 breeding pairs descendants list, so the total needs to be reduced by 4*(4-1)=12. And finally, there are 16 2nd cousins who would appear in the descendants list of 2 breeding pairs, and so the total needs to be reduced by 16*(2-1)=16. This leads to a correct tally of 128-7-7-12-16=86.
2. The calculation does not take into consideration the fact that those descendant will be 3rd cousins OR CLOSER.
Of the 86 in the above calculation, not all of them will be exactly 3rd cousins. Some of them will be closer (the person's sibling, for an obvious example).
Of the 86 total descendants of 8 breeding pairs leading to families to include relations to 3rd cousins, the breakdown is this:
64 3rd cousins
16 2nd cousin
4 1st cousins
1 sibling
1 self
=86
3. The calculation given prior only applies to people in the terminal generation. But there are other people who would exist who are related to within 3rd cousinship. An obvious example would be an aunt or uncle, but also should include all 2nd cousins, once removed, and first cousins, once removed, etc.
So it is not nearly so simple as just applying an exponential times the numer of breeding pairs.
