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The combination formula is a fundamental concept in combinatorics, used to calculate the number of ways to choose a subset of items from a larger set, without considering the order of selection. This is often represented with the notation (_{n}C_{r}), where 'n' is the total number of items to choose from, and 'r' is the number of items chosen.

The formula is given by:
\[ _{n}C_{r} = \frac{n!}{(n-r)! r!} \]
Here, the exclamation point denotes the factorial function, another key concept in mathematical expressions. To comprehend the combination formula deeply, let's use a practical example: If we have a set of 20 items (_{20}C_{5}) and wish to select 5, the formula helps us determine in how many unique ways this can be done, without repeating the same combination multiple times.

The factorial function is denoted by an exclamation mark '!' and is pivotal for evaluating expressions in combinatorics such as combinations and permutations. Essentially, the factorial of a non-negative integer 'n' is the product of all positive integers less than or equal to 'n'.

Mathematically, it's expressed as:
\[ n! = n \times (n-1) \times (n-2) \times ... \times 1 \]
For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). A special case is \( 0! \), which is defined to be 1. This function grows very rapidly with larger values of 'n', often requiring computational tools to evaluate expressions involving large factorials.

Evaluating expressions in mathematics typically involves substituting numbers for variables and performing the operations as indicated by the expression. In the context of permutations and combinations, this often entails plugging in values for 'n' and 'r' into the combination or permutation formulas and simplifying by calculating factorials.

For example, to evaluate the expression for (_{20}C_{5}), one would substitute 20 for 'n' and 5 for 'r', and then simplify: \[ 20! / (15! \times 5!) \]. The simplification process includes calculating the factorials and then dividing accordingly to get the final result, which gives the number of distinct combinations possible.

Permutations and combinations are two cornerstone concepts in combinatorics, each dealing with the arrangement of a set of objects. While permutations are concerned with the arrangement of all or part of a set of objects, with attention to the order, combinations are concerned with the selection of all or part of a set of objects, without attention to the order.

To differentiate, consider a set of objects 'A, B, C'. In permutations, the arrangement 'ABC' is different from 'CBA'. However, in combinations, these arrangements are considered the same because they contain the same objects irrespective of the order. Understanding this distinction is essential, as it greatly affects how we calculate the number of different possible arrangements or selections.