∑ Combinatorics Helpers

Binomial Expansion

Expand algebraic expressions and calculate specific terms step-by-step. Perfect for high school algebra.

Expression Terms

Supports positive integers up to 20.
Expression Preview

Common Binomial Presets

Basic Cubic Expansion
(x + y)3
Negative Terms & Coeffs
(2x − 3y)4
Find Coefficient of x2
In (x + 2)5
Find Middle Term
In (x + y)4
Expansion Solution
Step-by-Step Expansion Derivation

Introduction to the Binomial Theorem

The Binomial Theorem is a fundamental algebraic formula that provides a direct method for expanding any positive integer power of a binomial expression. It is written as:

(x + y)^n = ∑r=0n &binom;nr xn-r yr

where &binom;nr (often written as nCr) is the binomial coefficient, which represents the number of ways to choose r items from a set of n items.

Pascal's Triangle and Coefficients

The binomial coefficients for consecutive exponents can be neatly represented as rows in Pascal's Triangle, where each entry is the sum of the two numbers directly above it.

Mathematically, the coefficients are calculated using factorials:

&binom;nr = n! / [r! × (n - r)!]

Important Properties of Expansion

  • Number of Terms: The expansion of (x + y)n contains exactly n + 1 terms.
  • Powers Sum: In every single term, the sum of the exponents of the two variables is always equal to n.
  • Middle Terms:
    • If n is even, there is exactly one middle term, which is the (n/2 + 1)-th term.
    • If n is odd, there are two middle terms, which are the ((n+1)/2)-th and ((n+3)/2)-th terms.

Frequently Asked Questions

What is the general term in a binomial expansion? +
The general term (representing the (r+1)-th term) in the expansion of (a + b)n is given by the formula: Tr+1 = nCr · an−r · br. Note that because indices start at 0, the 5th term is represented by r = 4.
How does the expansion change with negative terms? +
If the expression contains a negative sign, e.g., (x − y)n, the terms will alternate signs. Terms with odd powers of the second variable (odd values of r) will be negative, while terms with even powers of the second variable (even values of r) will remain positive.
Can the exponent be negative or fractional? +
Yes, but the expansion is different. For negative or fractional exponents, we use the general binomial series: (1 + x)n = 1 + nx + n(n-1)x2/2! + .... Unlike positive integers, this expansion contains an infinite number of terms and is valid only if |x| < 1.