In answering this question we’ll take “k” to be the average number of children that a couple will have who survive to adulthood and themselves reproduce.
For the sake of the exercise, let’s assume that k = 2, with each generation just replacing itself with the next generation, and no overall increase in population.
In fact the population grows, so k=2 will give us a figure for 10th cousins at the lower end of the amount it is possible to have.
Assuming couple A has k children, and that those children themselves each have k children, then k squared (k2) first cousins or closer relations will exist.
After some generations k11 10th cousins or closer will exist.
If you look at one of your ancestral couples 10 generations back, you will have k11 10th cousins or closer through them.
But they are only one of your many ancestral couples at that time. You also have 10th cousins through your other ancestral couples.
Consider first cousins: you have two parents. Your first cousins are the descendants of your grandparents, minus your siblings.
Through each of the two grandparent couples, you get k2 first cousins, minus k for you and your siblings.
Total = 2 (k2 - k)
= 2 k (k-1)
Consider then second cousins: through each of the four relevant couples, you get k3 second cousins, minus k2 for your first cousins and closer.
Total = 4 (k3-k2)
= 22 (k2) (k-1)
= (2k)2 (k-1)
So your number of nth cousins will be = (2k)n (k-1)
And your number of 10th cousins = (2k)10 (k-1)
Assuming that k = 2, then the number of your 10th cousins is 410 = roughly a million (actually 1,048,576).
The number of 15th cousins is then 415 = roughly a billion (1,073,741,824).
But let’s say k is higher, and assume that an average three per generation survive and have children (a growing population).
Then your number of 10th cousins is 610 x 2 = 121 million.
And the number of 15th cousins is 615 x 2 = 940 billion (much larger than the population of the world – these are not, of course, necessarily all alive at the same time, and many are the product of intermarriage, doubling them up).
If k = 2.3, a reasonable assumption for the past, the numbers of 10th cousins rises above a million, with a bit of intermarriage then reducing their numbers again.
It is likely then that the answer to the question as to the number of 10th cousins we all have, is roughly one million.
MoreCreate your own family tree
Constructing your own family tree is made simple by programmes that take the hard work out of lining up the generations and connecting the cousins. You can do that by going online and using a simple-to-use free Family Echo / Irish Times programme at familyecho.com
Simply put in the details of a family member and then follow the programme's prompts and fill in the connections to parents, siblings and other relations. It will draw the tree. You can save the tree by registering on the site and amend or expand it at any stage. The tree can be downloaded or printed out.
Make sure you adjust the settings for paper size and for the number of generations you wish to show. It may also be necessary to print off several charts as the tree-builder will do in-laws separately - to focus on any one branch of the family just click on the key relation.
Take a look at an example of a family tree »
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