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I am using the binomial coefficient to calculate the number of 2-outcomes from $n$ items i.e. $\binom{n}{2}$ combinations.

I am looking for a way to write the list of combinations compactly in terms of subscript indices as they will form a $\binom{n}{2}$ length vector e.g.

If we have $n = 4$ items, $\binom{4}{2} = 6$, and the vector of elements is $\boldsymbol{\it x} = [x_{12},x_{13},x_{14},x_{23},x_{24},x_{34}]$.

The best I can come up with for the general $n$ item case is

$\boldsymbol{\it x} = [x_{12},x_{13},\ldots,x_{1n},x_{23},x_{24},\ldots,x_{2n},\ldots,x_{(n-1)n}]$

The audience for this particular work is those with a mechanical engineering background (which is my own field too) so I would like this notation to be clear while still retaining the general $n$ item description. Does my notation sufficiently explain the series of combinations or is there perhaps a more concise and/or clearer way to denote this?

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  • $\begingroup$ It rather depends on whether order matters. You could have something like $\{x_{i,j}\}_{1 \le i \lt j \le n}$ $\endgroup$ – Henry Feb 17 '14 at 15:43
  • $\begingroup$ Order doesn't matter, your notation is exactly the kind of thing I was looking for. If you submit that as an answer I'd be happy to accept that. $\endgroup$ – allie Feb 17 '14 at 17:22
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It rather depends on whether order matters. You could have something like $\{x_{i,j}\}_{1 \le i \lt j \le n}$ or $\{x_{i,j} : 1 \le i \lt j \le n\}$

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