# 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 ${\left(a+b\right)}^{n}$, where 'n' is a non-negative 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:

1. 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.'

2. Use the Binomial Theorem: Apply the binomial theorem formula to find the coefficient. The general formula is:

${\left(a+b\right)}^{n}$ = C(n, 0) * ${a}^{n}$ * ${b}^{0}$ + C(n, 1) * ${a}^{\left(n-1\right)}$) * ${b}^{1}$ + C(n, 2) * ${a}^{\left(n-1\right)}$ * ${b}^{2}$ + ... + C(n, n) * ${a}^{0}$ * ${b}^{n}$

• 'n' is the non-negative 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.
3. 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 ${\left(x+y\right)}^{5}$, 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:

${\left(x+y\right)}^{5}$ = 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 ${\left(2a-3b\right)}^{4}$, 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:

${\left(2a-3b\right)}^{4}$ = C(4, 0) *${\left(2a\right)}^{4}$ * ${\left(-3b\right)}^{0}$ + C(4, 1) * ${\left(2a\right)}^{3}$ * ${\left(-3b\right)}^{1}$ + C(4, 2) * ${\left(2a\right)}^{2}$ * ${\left(-3b\right)}^{2}$ + C(4, 3) * ${\left(2a\right)}^{1}$ * ${\left(-3b\right)}^{3}$ + C(4, 4) * ${\left(2a\right)}^{0}$ * ${\left(-3b\right)}^{4}$

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