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The binomial coefficient is a numerical value that shows how many combinations of elements can be picked from a larger set. In mathematics, it's often represented as \( \binom{n}{r} \), pronounced "n choose r". To calculate it, use the formula:

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

Here, \(n\) is the total number of items, and \(r\) is the number of items to choose. This coefficient is essential in binomial expansions because it determines the number of ways to pick terms.

For instance, if we want to find \( ^{10}C_{4} \), it would mean calculating how many ways we can choose 4 items from 10. Substituting into our formula, it becomes \( \frac{10!}{(10-4)! \, 4!} \), which simplifies to a specific integer after computing the factorial values.

A factorial, denoted by an exclamation mark (!) after a number, is the product of all positive integers up to that number. Factorials are fundamental in calculating combinations and permutations.

• For example, the factorial of 5 (written as \(5!\)) is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials grow very quickly as numbers increase. For practical applications like our binomial problem, factorials help determine the binomial coefficients by providing the total number of possible arrangements.

In the original exercise, calculating \( ^{10}C_{4} \) involves finding \(10!\), \(6!\), and \(4!\), each representing a factorial value used in binomial coefficient computation.

The binomial expansion formula is a method to expand expressions raised to a power, specifically, expressions of the form \((a + b)^n\). This formula allows you to expand such expressions into a sum of terms of the form \( \binom{n}{k} a^{n-k} b^{k} \).

The general term in a binomial expansion is given by:

• \( T_{k+1} = \binom{n}{k} a^{n-k} b^{k} \)

Here, \(n\) is the power to which the binomial is raised, \(k\) represents the term number minus one (hence \(k+1\) for the kth term), \(a\) is one part of the binomial, and \(b\) is the other.

In our exercise, for the expansion of \((x+8)^{10}\), we are seeking the 5th term. So, we calculate using \(k=4\) in the formula and substitute \(a = x\) and \(b = 8\).

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It's a shorthand for repeated multiplication of the base, written as \( a^n \). Here, \(a\) is the base and \(n\) is the exponent.

• For example, \(3^4\) means \(3 \times 3 \times 3 \times 3 = 81\).

Exponentiation plays a key role in the binomial theorem, where each term in an expansion is raised to certain powers. Powering elements ensures each term represents the multiplication of components, making up the whole expression product.

In the binomial expansion of \((x + 8)^{10}\), as seen in our problem, we calculate different terms by using specific powers for both \(x\) and \(8\), determined by their positions and the binomial coefficient.