Binomial Expansion Calculator
Expand (a + b)^n using the binomial theorem with BigInt support for large coefficients.
Enter an exponent n (0 to 100) to expand (a + b)^n
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. The expansion is given by the sum of terms involving binomial coefficients.
The Formula: (a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n
Binomial Coefficients: C(n, k) = n! / (k!(n-k)!) represents the number of ways to choose k items from n items. These coefficients form Pascal's Triangle, where each number is the sum of the two numbers directly above it.
Pascal's Triangle Connection: The coefficients in the expansion of (a + b)^n correspond to row n of Pascal's Triangle. For example, (a + b)^4 has coefficients 1, 4, 6, 4, 1, which is row 4 of the triangle.
Algorithm: This calculator computes binomial coefficients using an iterative approach that avoids computing large factorials directly. BigInt is used to handle arbitrarily large coefficients without overflow, allowing expansion of (a + b)^100 which has coefficients with over 29 digits.
Applications: The binomial theorem is fundamental in algebra, probability theory (binomial distribution), calculus (Taylor series), and combinatorics. It's used in polynomial expansion, approximation methods, and counting problems.
What is the binomial theorem?
The binomial theorem describes how to expand (a + b)^n. It states that (a + b)^n equals the sum of C(n,k) * a^(n-k) * b^k for k from 0 to n, where C(n,k) is the binomial coefficient 'n choose k'. This allows expanding any binomial raised to a power.
What is Pascal's Triangle?
Pascal's Triangle is a triangular array where each entry is the sum of the two entries above it. The nth row contains the binomial coefficients for expanding (a+b)^n. For example, row 4 is [1, 4, 6, 4, 1], giving (a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
What are binomial coefficients used for?
Binomial coefficients C(n,k) count the number of ways to choose k items from n items. They appear in probability (combinations), statistics (distributions), combinatorics, and algebra. They're also used in numerical analysis, approximation theory, and polynomial interpolation.
Can the binomial theorem work with negative or fractional exponents?
Yes, for negative or fractional exponents, the generalized binomial theorem produces an infinite series (not a finite polynomial). This was discovered by Newton and is fundamental in calculus. For example, (1+x)^(1/2) gives a series for square roots.