Exploring the Fifth Coefficient in the Expansion of (xy^{21})

The binomial theorem is a fundamental concept in algebra, providing a straightforward way to expand expressions of the form (a b)n. When the coefficients a and b are taken to be 1, and when x and y are variables, we can explore the coefficients in the expansion of (xy^{21}). This article delves into finding the fifth coefficient in the expansion using the binomial theorem, and provides a clear understanding of the underlying mathematics.

Understanding the Binomial Expansion

The binomial expansion of (a b)^n is given by:

(a b)^n Σk0nC(n, k) ? an-k ? bk

For (xy^{21}), we let (a x) and (b y), and find the fifth coefficient using (k 4).

Calculating the Fifth Coefficient

The fifth term in the expansion of (x y^{21}) can be identified using the binomial coefficient:

C(21, 4) ? x17 ? y4

This term is derived by setting k 4 because the first term is C(21, 0) ? x21 ? y0, and it follows the general structure of the binomial expansion.

The coefficient (C(21, 4)) can be calculated as:

C(21, 4) 21! / (4! ? 17!)

Which simplifies to:

(21 × 20 × 19 × 18) / (4 × 3 × 2 × 1) 5985

Binomial Expansion of (xy^{21})

The full expansion of (xy^{21}) is as follows:

First term: C(21, 0) ? x21 ? y0 1 ? x21 Second term: C(21, 1) ? x20 ? y1 Third term: C(21, 2) ? x19 ? y2 Fourth term: C(21, 3) ? x18 ? y3 (21 ? 20 ? 19 / 6) ? x18 ? y3 Fifth term: C(21, 4) ? x17 ? y4 5985 ? x17 ? y4 Etc., until the 22nd term: C(21, 21) ? x0 ? y21 1 ? y21

Further Insights and Applications

Finding the coefficients in the binomial expansion is not limited to academic interest. It plays a crucial role in various applications, such as data science, probability theory, and combinatorics. For a deeper dive into binomial coefficients and their computation, refer to resources like Khan Academy, Wikipedia, and YouTube tutorials on the binomial theorem and Pascal's triangle.

Conclusion

Understanding the fifth coefficient in the expansion of xy^{21} is just the tip of the iceberg when it comes to the binomial theorem. Exploring the full expansion and the underlying principles can unlock a wealth of mathematical insight. Whether for academic purposes or practical applications, the binomial theorem provides a robust framework for solving complex problems involving variable exponents.