Choose 2 numbers from 100 numbers

Question

If I want to choose a set of 2 positive integers less than 100, is that a permutation or a combination? I can't understand does the order matter in this case or it does not?

Answer 1

This would be a combination, since the order of the 100 items doesn't matter.

The number of ways to choose $k$ out of $n$ is given by the binomial coefficient \begin{pmatrix}n \\ k \end{pmatrix} which in your case would be $$\begin{pmatrix}100 \\ 2 \end{pmatrix}=4950$$

Edit: As amWhy points out below, it may be more appropriate to interpret OP's question as "2 out of 99 numbers". Either way, the formula holds.

Answer 2

Informally, a permutation of a set is a rearrangement of all its elements. That is, a subset with two elements is not a permutation, simply because it has only two elements. I think that you are messing up permutations and variations.

A variation with $2$ elements is an ordered pair. So the variations $42,18$ and $18,42$ are different.

A combination with two elements is a set, or more precisely, a subset. Two subsets are the same if and only if they have the same elements, no mattter the order in which you write them. So the combinations $\{18,42\}$ and $\{42,18\}$ are one and the same.

So there are more variations than combinations.

To distinguish them, an useful but somewhat dangerous question is "Does the order matter?". A safer question is: "Choosing the same elements in two different orders, does it yield different results?"