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In the binomial expansion, the binomial coefficient is an essential part of calculating each term. The binomial coefficient is represented as \(\binom{n}{k}\), and it tells you how many ways you can choose k elements from a set of n elements. Mathematically, it is defined as: \[\binom{n}{k} = \frac {n!} {k!(n-k)!} \] Here, \(!\) denotes the factorial of a number, which is the product of all positive integers up to that number. For example, \(!5 = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Each coefficient in the binomial expansion is calculated using this formula.

In binomial expansion, the general term helps you find any specific term in the expansion without having to expand the entire expression. The general term in the expansion of \((a + b)^n\) is represented as: \[T_{k+1} = \binom{n}{k} a^{n-k} b^k \] Here: \(\binom{n}{k}\) is the binomial coefficient \(a^{n-k}\) and \(b^k \) are the variables raised to appropriate powers based on the term number. To find the k\textsuperscript{th} term, replace \k\ with the value you are interested in and \ with the total power. For example, if you want to find the 3\textsuperscript{rd} term of \((a + b)^7\), you set \(k + 1 = 3 \) which gives \k = 2\. Substitute \ = 7\ and \k = 2\ into the formula to get \(\binom{7}{2} a^{7-2} b^2 = \binom{7}{2} a^5 b^2 \).

The Binomial Theorem provides a quick way to expand binomial expressions raised to any power. It states that: \[(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n} a^0 b^n\] This formula tells us that each term in the expansion comprises a binomial coefficient, powers of \a\, and powers of \b\. For example, to expand \((a + b)^3\) using the theorem: \[(a + b)^3 = \binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a^1 b^2 + \binom{3}{3} a^0 b^3\] Simplifying, you get: \[(a + b)^3 = 1 \times a^3 + 3 \times a^2 b + 3 \times a b^2 + 1 \times b^3 \] Summarizing, the Binomial Theorem helps you find any term or the entire expanded form of a binomial raised to a power.