# 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 $S$.

$$S=\{\{p_i,p_j\}\},1\leq i<j\leq 6$$

or

$$S=\{(p_i,p_j)\},1\leq i,j\leq 6,i\ne j$$?

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Does picking $(p_1,p_2)$ generate a different result than picking $(p_2,p_1)$? – Thomas Andrews Nov 1 '12 at 16:14
If it are "pairs" i guess yes. – Epsilon Nov 1 '12 at 16:15
No, I think not. In this context (both win same prize), if, say person $p_1 \;\text{and } p_2$ are both chosen, that's no different that saying $p_2 \;\text{and } p_1$ are both chosen, i.e., the pair $(p_1, p_2)$ is no different than the pair $(p_2, p_1)$. If the first person chosen were to win a grand prize, and the second were to win a lesser/different prize, then order would matter, and you'd want to list both pairs. – amWhy Nov 1 '12 at 16:16
So if I pick $(p_1,p_2)$ as a pair to win a prize, the result is different than if I pick $(p_2,p_1)$ to win a prize? If I give \$10 each to Joe and Jane, is that different than if I give \$10 to each of Jane and Joe? – Thomas Andrews Nov 1 '12 at 16:18
Then, as I mention in my comment above, since order doesn't matter, you need only list $\{p_1, p_2\}$. Can you solve the problem with that issue cleared up? Think of two slots (one slot for each winner): How many possibilities are there for the first slot? When that first slot is filled, then how many possibilities remain for the second slot? – amWhy Nov 1 '12 at 16:24

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.