The exact coefficient of $x^6$ in the expansion of $(2+2x)^{10}$ is 215,040.
Understanding Binomial Expansion
To determine the coefficient of a specific term within a binomial expression raised to a power, we utilize the Binomial Theorem. This fundamental theorem provides a systematic way to expand expressions of the form $(a+b)^n$ without requiring tedious direct multiplication.
The general formula for any term (the $(k+1)$-th term) in the binomial expansion of $(a+b)^n$ is:
$T_{k+1} = \binom{n}{k} a^{n-k} b^k$
Where:
- $n$ represents the power to which the binomial is raised.
- $k$ is the index of the term, starting from $k=0$ for the first term.
- $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.
- $a$ is the first term of the binomial.
- $b$ is the second term of the binomial.
Identifying Components for $(2+2x)^{10}$
For the given expression, $(2+2x)^{10}$, we can identify the following components:
- $a = 2$
- $b = 2x$
- $n = 10$
We are specifically looking for the coefficient of the $x^6$ term. In the general term $\binom{n}{k} a^{n-k} b^k$, the power of $x$ originates from $b^k$. Since $b = 2x$, we have $b^k = (2x)^k = 2^k x^k$. For the term to contain $x^6$, we must have $x^k = x^6$, which implies $k=6$.
Calculating the Coefficient of $x^6$
With $k=6$, we can substitute these values into the binomial term formula:
$T_{6+1} = T_7 = \binom{10}{6} (2)^{10-6} (2x)^6$
This expression simplifies to:
$T_7 = \binom{10}{6} (2)^4 (2^6 x^6)$
The coefficient of $x^6$ is the entire numerical part of this term: $\binom{10}{6} \cdot 2^4 \cdot 2^6$.
Step-by-Step Calculation
The calculation involves two main parts: the binomial coefficient and the powers of 2. It is known that the coefficient of $x^6$ in $(2+2x)^{10}$ is derived from the product of $\binom{10}{6}$, $2^4$, and $2^6$.
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Calculate the Binomial Coefficient ($\binom{10}{6}$):
$\binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!}$
$= \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}$
$= 10 \times 3 \times 7$ (after simplifying the numerator and denominator)
$= 210$ -
Calculate the Powers of 2:
From $(2)^{10-6}$, we get $2^4 = 16$.
From $(2x)^6$, the constant part is $2^6 = 64$.
Multiplying these two powers of 2 together gives $2^4 \cdot 2^6 = 2^{4+6} = 2^{10}$.
$2^{10} = 1024$. -
Multiply to find the Final Coefficient:
The coefficient is the product of the binomial coefficient and the combined power of 2:
Coefficient = $\binom{10}{6} \times 2^{10} = 210 \times 1024$.
The detailed calculation is summarized in the table below:
| Factor | Value |
|---|---|
| $\binom{10}{6}$ | 210 |
| $2^{10}$ | 1024 |
| Total | 215,040 |
Therefore, the exact coefficient of $x^6$ in the expansion of $(2+2x)^{10}$ is 215,040.
Practical Applications of Binomial Expansion
Understanding binomial expansion is a fundamental concept with wide-ranging applications:
- Probability Theory: It forms the basis for the binomial probability distribution, used to model the number of successes in a fixed number of independent Bernoulli trials.
- Financial Mathematics: Used in option pricing models and other areas where compound growth is analyzed.
- Computer Science: Applied in algorithms and data structures, especially in combinatorics and complexity analysis.
- Engineering and Physics: Used in various formulas and models to simplify complex expressions.
The method demonstrated here is a versatile tool for finding coefficients of specific terms in any binomial expansion, providing a structured approach to solving such problems.
