Coefficient of a Particular Term
Coefficient of a Term in the Binomial Expansion

The binomial theorem provides a systematic way to expand expressions of the form ${(a+b)}^{n}$, where 'n' is a nonnegative integer and 'a' and 'b' are any real or complex numbers.

The expansion consists of multiple terms, each of which includes a coefficient and powers of 'a' and 'b.'

The coefficient of a particular term represents the constant factor that multiplies that specific term in the expanded expression.
Finding the Coefficient of a Specific Term
To find the coefficient of a particular term in the binomial expansion, follow these steps:

Identify the Term: Determine which term in the binomial expansion you want to find the coefficient for. Specify the term using its exponents of 'a' and 'b.'

Use the Binomial Theorem: Apply the binomial theorem formula to find the coefficient. The general formula is:
${(a+b)}^{n}$ = C(n, 0) * ${a}^{n}$ * ${b}^{0}$ + C(n, 1) * ${a}^{(n1)}$) * ${b}^{1}$ + C(n, 2) * ${a}^{(n1)}$ * ${b}^{2}$ + ... + C(n, n) * ${a}^{0}$ * ${b}^{n}$
 'n' is the nonnegative integer.
 'C(n, k)' represents the binomial coefficient, which is used to find the coefficient of a specific term.
 'a' and 'b' are the constants.

Identify the Coefficient: After applying the binomial theorem formula, you can find the coefficient of the term you're interested in.
Example 1:
Given the binomial expansion ${(x+y)}^{5}$, find the coefficient of the term with '${x}^{2}$ * ${y}^{3}$.'

The term you're interested in is '${x}^{2}$ * ${y}^{3}$.'

Apply the binomial theorem formula:
${(x+y)}^{5}$ = C(5, 0) * ${x}^{5}$ * ${y}^{0}$ + C(5, 1) * ${x}^{4}$ * ${y}^{1}$ + C(5, 2) * ${x}^{3}$ * ${y}^{2}$ + C(5, 3) * ${x}^{2}$ * ${y}^{3}$ + C(5, 4) * ${x}^{1}$ * ${y}^{4}$ + C(5, 5) * ${x}^{0}$ * ${y}^{5}$

The coefficient of the term '${x}^{2}$ * ${y}^{3}$' is C(5, 3) = 10.
Example 2:
In the binomial expansion ${(2a3b)}^{4}$, find the coefficient of the term with '${a}^{2}$ * ${b}^{2}$.'

The term you're interested in is '${a}^{2}$ * ${b}^{2}$.'

Apply the binomial theorem formula:
${(2a3b)}^{4}$ = C(4, 0) *${\left(2a\right)}^{4}$ * ${(3b)}^{0}$ + C(4, 1) * ${\left(2a\right)}^{3}$ * ${(3b)}^{1}$ + C(4, 2) * ${\left(2a\right)}^{2}$ * ${(3b)}^{2}$ + C(4, 3) * ${\left(2a\right)}^{1}$ * ${(3b)}^{3}$ + C(4, 4) * ${\left(2a\right)}^{0}$ * ${(3b)}^{4}$

The coefficient of the term '${a}^{2}$ * ${b}^{2}$' is C(4, 2) = 6.