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Binomial expansion is a way of expanding expressions that are raised to a power. When you see something like \((a+b)^n\), the binomial expansion enables you to express it as a sum of terms involving powers of \(a\) and \(b\). This is very handy because instead of multiplying \(a+b\) by itself \(n\) times, you can use the binomial theorem to directly calculate terms of the form \( \binom{n}{k} a^{n-k} b^k \).The binomial theorem formula is:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Here, \( \binom{n}{k} \) are called binomial coefficients, which tell you how many ways you can choose \(k\) elements out of \(n\). You can use this expansion to find any specific term without expanding the entire series, a significant time-saver especially for high powers like \(n=25\). In our exercise, the goal was to identify the first three terms of this expansion without calculating the entire series, using the binomial expansion approach.

Binomial coefficients are the building blocks of the binomial expansion, represented as \( \binom{n}{k} \), which is read as "n choose k." These coefficients indicate the number of ways to choose \(k\) elements from a set of \(n\) elements, and they are essential for finding specific terms in a binomial expansion.Mathematically, the binomial coefficient is calculated as follows:

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

where \(!\) denotes the factorial operation, which is the product of all positive integers up to that number.In our original exercise, the various chosen values of \(k\) (0, 1, 2) resulted in specific binomial coefficients for each of the first three terms:

• First term: \( \binom{25}{0} = 1 \)
• Second term: \( \binom{25}{1} = 25 \)
• Third term: \( \binom{25}{2} = 300 \)

This systematic usage of binomial coefficients simplifies the process of determining which terms appear in the expansion and their respective coefficients.

Understanding exponents is crucial in applying the binomial theorem, especially when finding terms in the expansion. In the expression \((a+b)^n\), every term is a product of different powers of \(a\) and \(b\).The general term of the expansion is given by \( \binom{n}{k} a^{n-k} b^k \). This reveals that the exponents on \(a\) and \(b\) always add up to \(n\). In our given problem, exponents played a key role, especially since the terms involved fractional powers such as \(c^{2/5}\) and \(c^{4/5}\):

• In the first term, \((3c^{2/5})^{25} = 3^{25} c^{10}\)
• In the second term, \((3c^{2/5})^{24} \times (c^{4/5})^1 = 3^{24} c^{49/5}\)
• In the third term, \((3c^{2/5})^{23} \times (c^{4/5})^2 = 3^{23} c^{54/5}\)

These calculations illustrate how exponents combine within the series to produce powers of the variable \(c\) for each term. Such combinations are integral for capturing the correct degree of each term when working with binomial expansions.