How can combinations be represented mathematically?

Combinations can be represented mathematically using the formula for combinations, which calculates how many ways you can select a subset of items from a larger set where the order of selection does not matter. The correct expression for this is given by:

[

C(n, r) = \frac{n!}{(n-r)!r!}

]

In this formula, (n) represents the total number of items in the set, (r) represents the number of items to be chosen, (n!) (n factorial) represents the product of all positive integers up to (n), ((n-r)!) represents the factorial of the difference between the total items and the items chosen, and (r!) accounts for the arrangements of the items being chosen.

This formula divides the total arrangements of (n) items by the arrangements of the (r) items and the arrangements of the remaining items, effectively removing the effects of order among the chosen items.

Thus, the representation captures the essence of combinations accurately by reflecting the number of distinct groups that can be formed from the original set.