Combinatorics, listing all pairs In a group of six people $p_1,\ldots,p_6$, two people are chosen to win a prize (=holiday on Tahiti). List all pairs we can make. This is a sample space $S$.
Is the answer
$$S=\{\{p_i,p_j\}\},1\leq i<j\leq 6$$
or
$$S=\{(p_i,p_j)\},1\leq i,j\leq 6,i\ne j$$?
 A: The word "pair" is generally used to denote an unordered pair (i.e. a set of size two). Such a pair is usually written $\{p_1, p_2\}$ to emphasize that it is a set (order does not matter). The context here suggests that this is the intended meaning of "pair", since both chosen people appear to win the same prize.
If the problem were such that two different prizes were being awarded, then we would speak of "ordered pairs". Such ordered pairs are usually written $(p_1,p_2)$, which is different from $(p_2,p_1)$, perhaps with the convention that the big prize winner goes in the first slot, while the small prize winner goes in the second.

In terms of sets, you might write your answer as
$$
\{\{p_i, p_j\} \mid 1 \leq i < j \leq 6\}.
$$
Specifying $i < j$ in the makes certain that you do not list sets like $\{p_1, p_1\}$ (the same person winning twice) or $\{p_2, p_1\}$ (which should have already been counted as $\{p_1, p_2\}$).
It might also be worth noting that there are $\binom{6}{2} = 15$ such pairs, since forming the list is equivalent to finding all the ways to choose two indices out of six in which order does not matter.
