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Finding the coefficient of a particular term in a binomial expansion is a vital skill in algebra. In our specific example, we aim to identify the coefficient for the term in the expansion

• Given: \((r - s^2)^{10}\)
• Target Term: \(r^6 s^8\)

The coefficient is determined by several factors in binomial expansion:

• The binomial coefficient \( \binom{n}{k} \) obtained using combinatorics
• The sign generated by the components

In this case, we find that the binomial coefficient is \( \binom{10}{4} = 210 \). Since \( (-s^2)^4 \) results in a positive product due to the even power, the final coefficient remains positive. Thus, the coefficient \( C \) for the term \( r^6 s^8 \) is given by 210.

The Binomial Expansion is a key concept that allows us to expand and express powers of binomial expressions. Using the binomial theorem, \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\],we can break down a binomial expression into individual terms.In our example \((r - s^2)^{10}\), we substitute:

• \(a = r\)
• \(b = -s^2\)
• \(n = 10\)

The theorem guides us in calculating each term by selecting appropriate values of \(k\). These values indicate the particular combinations and powers of \(r\) and \(s\) in the expanded terms. By matching powers to target \(r^6 s^8\), we specifically select \(k = 4\), ensuring both conditions for powers align properly.

Combinatorics plays a crucial role in binomial expansion, particularly when calculating coefficients. Combinatorial coefficients, or **binomial coefficients**, are derived from choosing elements, signifying combinations possible given a set number of items.For instance, the coefficient \(\binom{10}{4}\) originates from observing the number of ways to select 4 items from a total of 10 and is calculated as:\[\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\]Each combination of items contributes to forming specific terms in binomial expansion. In our case, this combinatorial principle enabled us to discover that the coefficient of \( r^6 s^8 \) stood at 210, showcasing the efficient application of combinatorics in algebra.