What’s the way to express the following series: 1,3,6,10…
I’m looking at a spreadsheet that shows migration to and from counties. There are ~3,000 counties, but how many relationships do I need to end up with. I’m only interested in showing one pair for each relationship, i.e., from County A to B, I only want A:B as a case (with the case having variables for both directions); I don’t need A:B, B:A. Make sense? So, for a three county nation: A:B, A:C, C:B. If I add D, I just add A:D, B:D, C:D, for 6 cases. How is this expressed in math terms? Essentially, I’m just adding up all the numbers up to, but not including n. If I have 3,000 counties, what is the formula? Thanks.
The answer is called the number of combinations of 3000 things taken 2 at a time., usually pronounced “3000 choose 2”. The formula you want is 3000!/2!(3000-2)!
If you try that on a calculator, then answer will be a mess since 3000! is so huge. particularly for a choose number as small as 2, better to do it algebraically.
It turns into 3000 * 2999 / 2 after you factor out the numbers common to the numerator and denominator.
For the special case of choosing 2 items, ie a pariwise selection, then what Giles beat me to is also 100% true, although for a different reason.
Thanks. However, the resulting number (~4.5million) seems too high. Maybe the spreadsheet I have has left out counties, where there were no moves back and forth. That would make sense, but the result is still only a bit over 1 million cases, not three. The tables I’m looking at are at the bottom of this Census Bureau page: http://www.census.gov/population/www/cen2000/ctytoctyflow.html
