# Expand (a + b)^n for Positive Integer n

# Expand (a + b)^n for Positive Integer n

**Key Concepts**

- To expand an expression like (a + b)^n, we generally use the
**binomial theorem**. - The expansion involves understanding and using the
**binomial coefficients**, which can be found in Pascal’s Triangle. These coefficients can also be calculated using the combination formula. **Positive integer n**refers to the exponent of the binomial.

**Steps to Expansion**

- Use the formula for the Binomial Theorem, which is (a + b)^n = Σ (nCk) (a^(n-k)) (b^k) from k = 0 to n.
- The term
**nCk**represents the coefficients in the expansion. It can be calculated as n! / (k!(n-k)!). - Expand the binomial (a + b)^n term by term.
- Possible values of k range from 0 to n. For each term, decrease the power of a by one and increase the power of b by one.

**For example**

Let’s consider expanding (a + b)^3

- When n=3, we consider the fourth row of Pascal’s Triangle, which is 1, 3, 3, 1.
- Using the formula and coefficients, we get: (a + b)^3 = 1
*a^3*b^0 + 3*a^2*b^1 + 3*a^1*b^2 + 1*a^0*b^3 - Simplifying this gives us: a^3 + 3a^2b + 3ab^2 + b^3.

**Practical Tips**

- Remember, the sum of the indices of ‘a’ and ‘b’ in each term is equal to the original power ‘n’.
- For larger values of ‘n’, using the binomial theorem along with Pascal’s Triangle or the combination formula is less error-prone than multiplying (a + b) repeatedly.