Binomial Series with k Binomial Coefficient

The binomial series is a way to express the expansion of the power of a binomial expression. A binomial expression is one that contains two terms, such as $(a + b)$. The binomial theorem provides a formula for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer.

Binomial Theorem

The binomial theorem 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, also known as "n choose k," and it is given by:

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

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

Binomial Coefficient with a Multiplicative Factor

Sometimes, we encounter a binomial coefficient multiplied by a constant factor $k$. This factor can be incorporated into the binomial series expansion. The modified term in the series would look like:

$$k \cdot \binom{n}{k} a^{n-k} b^k$$

Table of Differences and Important Points

Aspect	Binomial Coefficient	k Binomial Coefficient
Definition	$\binom{n}{k}$ is the number of ways to choose $k$ elements from a set of $n$ distinct elements.	$k \cdot \binom{n}{k}$ is the binomial coefficient multiplied by a constant factor $k$.
Formula	$\binom{n}{k} = \frac{n!}{k!(n-k)!}$	$k \cdot \binom{n}{k} = k \cdot \frac{n!}{k!(n-k)!}$
Role in Binomial Series	Coefficient of the term $a^{n-k} b^k$ in the expansion of $(a + b)^n$.	Coefficient of the term $a^{n-k} b^k$ in the modified binomial series.
Example Expansion Term	$\binom{5}{2} a^3 b^2$	$3 \cdot \binom{5}{2} a^3 b^2$

Examples

Example 1: Binomial Coefficient in Expansion

Consider the expansion of $(a + b)^5$. Using the binomial theorem, we have:

$$(a + b)^5 = \sum_{k=0}^{5} \binom{5}{k} a^{5-k} b^k$$

The terms in the expansion are:

$$\binom{5}{0} a^5 b^0 + \binom{5}{1} a^4 b^1 + \binom{5}{2} a^3 b^2 + \binom{5}{3} a^2 b^3 + \binom{5}{4} a^1 b^4 + \binom{5}{5} a^0 b^5$$

Example 2: k Binomial Coefficient in Expansion

Now, let's consider a modified expansion where each binomial coefficient is multiplied by its corresponding $k$:

$$(a + b)^5 = \sum_{k=0}^{5} k \cdot \binom{5}{k} a^{5-k} b^k$$

The terms in the modified expansion are:

$$0 \cdot \binom{5}{0} a^5 b^0 + 1 \cdot \binom{5}{1} a^4 b^1 + 2 \cdot \binom{5}{2} a^3 b^2 + 3 \cdot \binom{5}{3} a^2 b^3 + 4 \cdot \binom{5}{4} a^1 b^4 + 5 \cdot \binom{5}{5} a^0 b^5$$

Simplifying, we get:

$$a^5 + 5a^4b + 20a^3b^2 + 30a^2b^3 + 20ab^4 + 5b^5$$

Notice that the term with $k=0$ is always zero because it is multiplied by $0$.

Conclusion

The binomial series with a $k$ binomial coefficient introduces a multiplicative factor to each term of the binomial expansion. This factor can significantly alter the coefficients in the series, and it's important to account for it when performing expansions. The examples provided illustrate how the factor $k$ changes the coefficients in the binomial series.