What Is The Coefficient Of In The Expansion Of

What is the Coefficient of x^k in the Expansion of (1+x)ⁿ?

The binomial theorem is a fundamental concept in algebra, providing a powerful tool for expanding expressions of the form (a + b)ⁿ. Understanding how to extract specific coefficients from these expansions is crucial in various mathematical applications, from probability theory to combinatorics. This article delves into the specifics of finding the coefficient of x^k in the expansion of (1 + x)ⁿ, exploring the underlying principles and providing practical examples.

Understanding the Binomial Theorem

The binomial theorem states that for any non-negative integer n and any real numbers a and b:

(a + b)ⁿ = Σ (n choose k) * a^(n-k) * b^k

where the summation runs from k = 0 to n, and (n choose k) represents the binomial coefficient, often written as _nC_k, ⁿC_k, or C(n, k). This coefficient is calculated as:

(n choose k) = n! / (k! * (n-k)!)

where n! denotes the factorial of n (i.e., n! = n * (n-1) * (n-2) * ... * 2 * 1). The binomial coefficient represents the number of ways to choose k items from a set of n distinct items.

Focusing on (1 + x)ⁿ

In our specific case, we're interested in the expansion of (1 + x)ⁿ. Applying the binomial theorem, we get:

(1 + x)ⁿ = Σ (n choose k) * 1^(n-k) * x^k

Since 1^(n-k) always equals 1, this simplifies to:

(1 + x)ⁿ = Σ (n choose k) * x^k

This equation directly reveals that the coefficient of x^k in the expansion of (1 + x)ⁿ is simply (n choose k).

Calculating the Binomial Coefficient

Let's break down how to calculate (n choose k):

1. Factorials: First, we need to understand factorials. The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. For example:

   • 5! = 5 * 4 * 3 * 2 * 1 = 120
   • 0! = 1 (by convention)

2. The Formula: The binomial coefficient is calculated using the formula:

   (n choose k) = n! / (k! * (n-k)!)

3. Example: Let's find the coefficient of x³ in the expansion of (1 + x)⁵. Here, n = 5 and k = 3.

   (5 choose 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 10

Therefore, the coefficient of x³ in the expansion of (1 + x)⁵ is 10.

Pascal's Triangle and Binomial Coefficients

Pascal's Triangle provides a visually intuitive way to generate binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it. The nth row of Pascal's Triangle contains the coefficients of the expansion of (1 + x)ⁿ. For instance:

• Row 0: 1 ((1+x)⁰ = 1)
• Row 1: 1 1 ((1+x)¹ = 1 + x)
• Row 2: 1 2 1 ((1+x)² = 1 + 2x + x²)
• Row 3: 1 3 3 1 ((1+x)³ = 1 + 3x + 3x² + x³)
• Row 4: 1 4 6 4 1 ((1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴) and so on...

This visual representation reinforces the relationship between binomial coefficients and the expansion of (1 + x)ⁿ.

Applications and Significance

The ability to determine the coefficient of x^k in the expansion of (1 + x)ⁿ has widespread applications:

• Probability: In probability theory, binomial coefficients represent the probability of obtaining exactly k successes in n independent Bernoulli trials (experiments with two possible outcomes, success or failure).

• Combinatorics: Binomial coefficients are fundamental in combinatorics, counting the number of ways to choose subsets from a set. They are used to solve problems involving combinations and selections.

• Calculus: The binomial theorem plays a crucial role in calculus, particularly in binomial series expansions and Taylor series expansions of functions.

• Statistics: Binomial distributions, heavily reliant on binomial coefficients, are used extensively in statistical analysis to model the probability of discrete events.

Advanced Considerations: Negative and Fractional Exponents

While the binomial theorem is primarily defined for non-negative integer exponents, it can be extended to negative and fractional exponents using the generalized binomial theorem. However, this extension results in infinite series, and the concept of "coefficient" becomes more nuanced. The series converges only under certain conditions (|x| < 1 for negative and fractional exponents). For these cases, the formula for the coefficient of x^k becomes:

(n choose k) = n(n-1)...(n-k+1) / k!

It's crucial to note that for negative or fractional n, this generates an infinite series, meaning there is no finite number of terms.

Example using the generalized binomial theorem

Let's consider finding the coefficient of x³ in the expansion of (1 + x)^-2. Here, n = -2 and k = 3.

Using the generalized binomial theorem formula:

(-2 choose 3) = (-2)(-3)(-4) / (3!) = -24/6 = -4

Thus, the coefficient of x³ in the expansion of (1+x)^-2 is -4. Remember this is one term in an infinite series.

Conclusion

The coefficient of x^k in the expansion of (1 + x)ⁿ, for non-negative integer n, is given by the binomial coefficient (n choose k). This fundamental concept finds broad application in diverse mathematical fields, highlighting the importance of understanding its calculation and significance. While the concept extends to negative and fractional exponents through the generalized binomial theorem, it's essential to grasp the difference between finite expansions and infinite series resulting from this extension. Mastering the binomial theorem and its applications opens doors to a deeper understanding of many mathematical and scientific disciplines.