General Term in Binomial Theorem
The binomial theorem is a powerful mathematical concept used to expand expressions of the form ${(a+b)}^{n}$, where 'a' and 'b' are real numbers, and 'n' is a nonnegative integer. The binomial theorem provides a systematic way to find the expansion of such expressions. In this context, the "general term" refers to an individual term within the expanded expression. Let's delve into the general term of the binomial theorem.
The General Term Formula:
The general term of the binomial expansion of ${(a+b)}^{n}$ can be represented using the binomial coefficient, also known as "n choose k" or "C(n, k)." The formula for the general term is as follows:
T(k) = C(n, k) * (${a}^{(nk)}$) * (${b}^{k}$)
Where:
 T(k) is the kth term in the expanded expression.
 C(n, k) is the binomial coefficient, given by C(n, k) = n! / (k! * (nk)!), where 'n!' represents the factorial of 'n.'
 'a' and 'b' are the constants in the original expression (a + b).
 'n' is the exponent to which the binomial is raised.
 'k' is the index of the term you want to find in the expansion. It ranges from 0 to 'n,' and each term corresponds to a specific value of 'k.'
Key Points:

Binomial Coefficient (C(n, k)): The binomial coefficient determines the number of ways to choose 'k' elements from a set of 'n' elements without regard to order. It plays a crucial role in the binomial expansion.

Exponent Rules: In the general term formula, 'a' and 'b' are raised to powers of 'nk' and 'k,' respectively. This represents the contribution of each term to the overall expansion.

Term Index (k): The value of 'k' ranges from 0 to 'n.' For ${(a+b)}^{n}$, there are (n+1) terms in the expansion. Each term corresponds to a unique value of 'k.'

Coefficient: The binomial coefficient acts as a multiplier for each term and determines its contribution to the expansion.
Examples:
Let's consider a few examples to better understand the general term in the binomial theorem.

For (${(a+b)}^{4}$, the general terms are:
 T(0) = C(4, 0) * ${a}^{4}$ * ${b}^{0}$ = 1 * ${a}^{4}$ * 1 = ${a}^{4}$
 T(1) = C(4, 1) * ${a}^{3}$ * ${b}^{1}$ = 4 * ${a}^{3}$ * b
 T(2) = C(4, 2) * ${a}^{2}$ * ${b}^{2}$ = 6 * ${a}^{2}$* ${b}^{2}$
 T(3) = C(4, 3) * ${a}^{1}$ * ${b}^{3}$ = 4 * a * ${b}^{3}$
 T(4) = C(4, 4) * ${a}^{0}$ * ${b}^{4}$ = 1 * ${b}^{4}$

For ${(x+2y)}^{3}$, the general terms are:
 T(0) = C(3, 0) * ${x}^{3}$ * ${\left(2y\right)}^{0}$ = 1 * ${x}^{3}$ * 1 = ${x}^{3}$
 T(1) = C(3, 1) * ${x}^{2}$ * 2y = 3 * ${x}^{2}$ * 2y
 T(2) = C(3, 2) * ${x}^{1}$ * ${\left(2y\right)}^{2}$ = 3 * x * 4${y}^{2}$
 T(3) = C(3, 3) * ${x}^{0}$ * ${\left(2y\right)}^{3}$ = 1 * 1 * 8${y}^{3}$ = 8${y}^{3}$