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Combinations are ways to select items from a group, where the order of selection does not matter. This is a key concept in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of objects. When talking about combinations, we're often interested in finding the number of ways we can choose a subset of items from a larger set, regardless of the order.

To illustrate, imagine you're at a buffet with 10 different dishes, but you can only try 3 due to limited room on your plate. Combinations will help you figure out how many different trios of dishes you can select. In this scenario, the choice of salad, pizza, and pasta is the same as pasta, salad, and pizza; their order doesn't affect the selection. This concept is critical when solving problems like the one about selecting hymns from hymn books; the order in which the hymns are chosen is irrelevant to the choir director.

The factorial is denoted by an exclamation mark (!) and it represents the product of all positive integers up to a given number. For example, the factorial of 5, written as 5!, is calculated as 5 x 4 x 3 x 2 x 1 = 120. Factorials are fundamental to computing combinations and permutations because they are used to detail the total number of ways to arrange a set of objects.

Why is Factorial Important in Combinations?

In combinations, we divide the factorial of the number of total items by the product of the factorial of the number of items chosen and the factorial of the difference between the two. This technique eliminates the arrangements which we are not considering, as order doesn't matter in combinations. The problem with selecting hymns showcases this by dividing 25! by 23! when determining the number of ways to choose 2 hymns from 25.

Permutations deal with the arrangement of objects where the order is important. Unlike combinations, every change in order of the items results in a different permutation. For example, the arrangement 'ABC' is different from 'CAB'. To calculate permutations, we also use factorial notation.

Let's say we have to arrange 3 books. The possibilities include 'Book1 Book2 Book3', 'Book2 Book3 Book1', etc. If the number of books increases, say to 6 like the hymns in our exercise, the number of possible arrangements grows exponentially. Thus, permutations are more numerous than combinations when we're working with the same set and number of selections because each arrangement counts as a unique outcome. This highlights why the distinction between permutations and combinations is crucial, especially in scenarios requiring specific arrangements or sequences.