\({ }^{\mathrm{n}} \sum_{\mathrm{r}=0}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} 4^{\mathrm{r}}=\) (a) \(4^{\mathrm{n}}\) (b) \(5^{\mathrm{n}}\) (c) \(4^{-n}\) (d) \(5^{-n}\)

Combinatorics is a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.

One of the key aspects of combinatorics is understanding how to count without actually enumerating cases individually. This involves identifying the structure in a set of objects and then using this structure to determine the count. For example, when using the binomial theorem for expansion, we rely on combinatory logic to determine the coefficients of the terms - these coefficients arise from combinatoric considerations.

Permutations and combinations are foundational concepts in combinatorics that deal with the arrangement of sets of items. A permutation refers to an arrangement of elements in a definite order, whereas a combination refers to a selection of elements where the order does not matter.

Understanding permutations and combinations is vital in binomial expansion, since the coefficients found in the expansion correspond to the number of ways to choose a certain number of elements from a larger set, disregarding order. For instance, in the binomial theorem, coefficients are determined by the binomial coefficients (symbolically represented as 'n choose r' or Cr), which are essentially an application of combinations. These coefficients tell us how many ways we can select r objects from a set of n without regard to the order of selection.

Algebraic expansion is the process of removing parentheses from an algebraic expression and simplifying the result. The binomial theorem is one of the most important tools for algebraic expansion when it comes to binomials raised to a power.

The binomial theorem states that (1 + x)^n = C0 + C1x + C2x^2 + ... + Cnx^n, where Ckcorresponds to (n choose k). In other words, it's the sum of the products of the binomial coefficients and the corresponding powers of the variables involved.

Expanding binomials using this theorem allows for a quick and efficient way to multiply out expressions that would otherwise require much more tedious multiplication. In our exercise, the expansion (1 + 4)^n = 5^n is made immediate by recognizing that this is the sum of 1 raised to various powers (which is always 1) and 4 raised to complementary powers (with respect to n), all multiplied by the appropriate binomial coefficients.