AkCalculators

Binomial Calculator — binomial theorem, expansion and binomial probability distribution

The binomial theorem and the binomial distribution are two pillars of algebra and probability. The Binomial Calculator combines both: expand expressions like (a + b)^n using binomial coefficients, and compute probabilities for discrete random variables following a Binomial(n, p) distribution. Use Expansion Mode for algebraic practice and symbolic coefficients; use Probability Mode to compute P(X = k), cumulative probabilities, and summary statistics like mean and variance.

The binomial theorem (expansion)

The binomial theorem states that for any non-negative integer n:

(a + b)^n = Σ_{k=0}^n C(n,k) · a^{n−k} · b^k

Where C(n,k) (also written nCk or "n choose k") is the binomial coefficient, the number of ways to choose k items from n:

C(n,k) = n! / (k! (n−k)!)

Expansion Mode enumerates coefficients C(n,0), C(n,1), …, C(n,n) and displays the fully expanded polynomial by assembling terms with powers of a and b. For symbolic 'a' or 'b' (like x), the calculator prints the algebraic expression; for numeric a,b it evaluates numeric terms as well.

Computing binomial coefficients reliably

Factorials grow quickly, so direct factorial computation can overflow. The calculator computes C(n,k) using cancellation and iterative multiplicative formulas (product of k terms) and uses BigInt for integer-safe results when supported. For large n the calculator will present coefficients and — where appropriate — warn about very large numbers.

Binomial probability distribution

A binomial random variable X ~ Binomial(n, p) counts the number of successes in n independent Bernoulli trials each with success probability p. The point probability is:

P(X = k) = C(n,k) · p^k · (1 − p)^{n−k},  for k = 0,1,...,n

The mean (expected value) and variance of X are:

μ = n·p,   σ² = n·p·(1 − p)

Probability Mode computes point probabilities, sums them for cumulative queries (e.g., P(X ≤ k), P(a ≤ X ≤ b)), and provides μ and σ². For numerical stability with moderate-to-large n, computations sometimes use logarithms (log-space) to avoid underflow/overflow; this tool uses stable multiplication and summation techniques for typical n encountered in coursework.

Examples

1. Expansion: (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1. Coefficients are 1,4,6,4,1 (row 4 of Pascal’s triangle).
2. Probability: For n=10, p=0.3, P(X=3) = C(10,3)*0.3^3*0.7^7 ≈ 0.2668. The calculator shows the multiplication of combination and power terms step-by-step.
3. Cumulative: P(X ≤ 3) is the sum P(0)+P(1)+P(2)+P(3); the tool computes and shows the partial sums.

When to use this calculator

• Algebra practice: expand binomials and verify coefficients.
• Probability problems: compute exact binomial probabilities or cumulative probabilities for hypothesis testing or examples.
• Teaching: show relationships among coefficients (Pascal’s triangle), symmetry C(n,k) = C(n,n−k), and combinatorial interpretations.

Practical notes and edge cases

• Probability p must be in [0,1]. Extreme p (0 or 1) yields degenerate distributions.
• k must satisfy 0 ≤ k ≤ n; out-of-range queries return zero or explained behavior.
• Large n with many terms may produce extremely small probabilities — the tool rounds display but preserves internal precision for typical classroom values.

Conclusion

The Binomial Calculator is a compact, dual-purpose tool that supports algebraic expansion with coefficients and practical probability computations for Binomial(n,p). It’s suitable for learning, checking homework, and exploring combinatorial identities and distributional behavior.