The coefficient of \(x^{8}\) in the product \((x+1)(x+2)(x+3) \ldots\) \((\mathrm{x}+10)\) is (A) 1024 (B) 1300 (C) 1320 (D) 1360

Polynomial expansion refers to the process of multiplying out the terms of a polynomial expression and combining like terms. It's a fundamental concept in algebra that allows us to simplify expressions where a polynomial is raised to a power or multiplied by another polynomial.

An example of such expansion is the distributive property, commonly recalled as FOIL (First, Outer, Inner, Last) when dealing with binomials. This method can become cumbersome when expanding polynomials with more than two terms, which is why mathematicians have developed more systematic approaches like the binomial theorem and Pascal's triangle to efficiently handle larger expansions.

Each term in a polynomial expansion is the result of multiplying one term from each of the factors. In our exercise, to find the coefficient of a specific term like \(x^8\), we must identify all the possible ways the terms from multiple binomials can be multiplied to result in an \(x^8\) term and then calculate the sum of their coefficients.

Combination mathematics, also known as combinatorics, deals with the counting, arrangement, and combination of elements within a set. The most relevant aspect of combination mathematics to our exercise is the concept of 'combinations,' which helps to determine the number of ways a certain number of items can be selected from a larger set when the order of selection does not matter.

In the context of polynomial expansion, combinations are used to determine how many ways terms can be selected to form a particular power of \(x\). For example, to find the coefficient of \(x^8\) in a product of several binomials, we look at all the possible selections of terms that would produce \(x^8\) when multiplied together.

To illustrate, selecting the \(x\) term from eight different binomials and summing the constant terms leftover from the remaining binomials will give us one of the combinations contributing to the coefficient of \(x^8\). However, we must be careful to only include valid combinations that stay within the constraints of our expression, as noted in the exercise improvement advice.

The binomial theorem is a powerful tool in algebra that provides a quick way to expand binomials raised to a power without manually multiplying the expression many times. It states that \((a+b)^n\) can be expanded into a sum of terms of the form \(C(n, k) a^{n-k} b^k\), where \(C(n, k)\) are the binomial coefficients and \(k\) ranges from 0 to \(n\). These coefficients can be found in Pascal's triangle or calculated directly using the formula \(C(n, k) = \frac{n!}{k!(n-k)!}\).

The binomial theorem simplifies the process of polynomial expansion and is applicable when we need to expand expressions like \((x+1)^n\), a critical aspect when dealing with polynomial expressions involving multiple terms. In the context of our exercise, understanding the binomial theorem can help to pinpoint how each term in the expansion contributes to the overall coefficient we are looking for.

While we did not directly apply the binomial theorem in the step-by-step solution due to the product involving more than two terms, having a grasp of this theorem allows a better understanding of the structures within polynomial expansions and the resulting coefficients.