Understanding the Binomial Coefficient of a Specific Type of Term

The binomial coefficient is a fundamental concept in combinatorics and plays a crucial role in the binomial theorem. It is used to describe the number of ways to choose a subset of elements from a larger set, and it also determines the coefficients in the expansion of a binomial expression raised to a power.

Binomial Theorem (BT)

The binomial theorem provides a formula for the expansion of a binomial raised to any positive integer power. It states that:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

where $\binom{n}{k}$ is the binomial coefficient, which can be calculated using the formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$.

Binomial Coefficient

The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose $k$ elements from a set of $n$ distinct elements. It is also the coefficient of the term $a^{n-k}b^k$ in the binomial expansion of $(a + b)^n$.

Properties of Binomial Coefficients

• Symmetry: $\binom{n}{k} = \binom{n}{n-k}$
• Pascal's Rule: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
• Boundary Conditions: $\binom{n}{0} = \binom{n}{n} = 1$

Finding the Coefficient of a Specific Type of Term

To find the coefficient of a specific term in the binomial expansion, you need to identify the powers of $a$ and $b$ in that term. For example, to find the coefficient of the term $a^{n-k}b^k$ in the expansion of $(a + b)^n$, you would use the binomial coefficient $\binom{n}{k}$.

Example

Find the coefficient of $x^3y^2$ in the expansion of $(x + y)^5$.

Solution: The term $x^3y^2$ corresponds to $k = 2$ since the power of $y$ is 2. Therefore, the coefficient is $\binom{5}{2}$.

$$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$$

So, the coefficient of $x^3y^2$ in the expansion of $(x + y)^5$ is 10.

Table of Differences and Important Points

Property	Description	Example
Notation	$\binom{n}{k}$ denotes the binomial coefficient.	$\binom{5}{2}$
Formula	$\binom{n}{k} = \frac{n!}{k!(n-k)!}$	$\binom{5}{2} = \frac{5!}{2!3!}$
Symmetry	The coefficient is symmetric: $\binom{n}{k} = \binom{n}{n-k}$	$\binom{5}{2} = \binom{5}{3}$
Pascal's Rule	Relates to adjacent coefficients: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$	$\binom{5}{2} = \binom{4}{1} + \binom{4}{2}$
Boundary	Always equals 1 when $k=0$ or $k=n$: $\binom{n}{0} = \binom{n}{n} = 1$	$\binom{5}{0} = \binom{5}{5} = 1$

Conclusion

Understanding the binomial coefficient is essential for working with binomial expansions and combinatorial problems. By recognizing the patterns and properties of binomial coefficients, one can efficiently determine the coefficients of specific terms in binomial expansions and solve a variety of mathematical problems.