How to Find the Coefficient of (x^9) in the Expansion of (x^{14} cdot 5^{14})

In this article, we will delve into the process of finding the coefficient of (x^9) in the expansion of (x^{14} cdot 5^{14}). This involves the application of the Binomial Theorem, a powerful mathematical tool for expanding binomials to any power. We will walk through the steps with detailed explanations and calculations.

Introduction to the Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra and combinatorics that allows us to expand expressions of the form ((a b)^n) into a sum of terms, each of which is a product of a binomial coefficient and powers of (a) and (b). The general form is:

[ (a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k ]

Applying the Binomial Theorem to (x^{14} cdot 5^{14})

Let's apply the Binomial Theorem to the given expression, (x^{14} cdot 5^{14}). Here:

[ a x ] [ b 5 ] [ n 14 ]

The general term in the expansion can be expressed as:

( binom{14}{k} x^{14-k} 5^k )

Identifying the Term Containing (x^9)

To find the term containing (x^9), we need to set [ 14 - k 9 ]. Solving this equation gives us:

[ k 5 ]

Calculating the Coefficient

Substituting (k 5) into the general term, we get:

[ binom{14}{5} x^{9} 5^5 ]

Binomial Coefficient Calculation

The binomial coefficient is given by:

[ binom{14}{5} frac{14!}{5!9!} ]

Calculating the factorials, we have:

[ frac{14 times 13 times 12 times 11 times 10}{5 times 4 times 3 times 2 times 1} frac{240240}{120} 2002 ]

Power of 5 Calculation

Calculating the power of 5, we get:

[ 5^5 3125 ]

Final Coefficient Calculation

The coefficient of (x^9) is then given by:

[ 2002 times 3125 6256250 ]

Thus, the coefficient of (x^9) in the expansion of (x^{14} cdot 5^{14}) is 6256250.

Conclusion

In this article, we have demonstrated a systematic approach to finding the coefficient of a specific term in a binomial expansion using the Binomial Theorem. By following these steps, you can tackle similar problems with confidence and precision. Understanding and applying the Binomial Theorem is a valuable skill, applicable in various fields including mathematics, physics, and engineering.