# Math ?: sorta like factorial **URL:** \ **Category:** Factual Questions\ **Created:** [March 19, 2005, 1:37pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309 "2005-03-19T13:37:53Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [March 19, 2005, 1:37pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309/1 "2005-03-19T13:37:53Z")
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. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [March 19, 2005, 1:42pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309/2 "2005-03-19T13:42:52Z")
…and no, this is not a homework assignment I’m cheating on. 🙂 ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [March 19, 2005, 1:47pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309/3 "2005-03-19T13:47:23Z")
n \* (n-1) / 2 The series is the triangular numbers: 1 is \* 3 is 1+2 \* \*\* 6 is 1+2+3 \* \*\* * * * 10 is is 1+2+3+4 \* \*\* * * * * * * etc. These numbers also appear in a diagonal line on Pascal’s triangle (the third column below):; 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [March 19, 2005, 1:53pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309/4 "2005-03-19T13:53:21Z")
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. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [March 19, 2005, 2:48pm UTC](https://boards.straightdope.com/t/math-sorta-like-factorial/295309/5 "2005-03-19T14:48:41Z")
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](http://www.census.gov/population/www/cen2000/ctytoctyflow.html) Thanks.