Last updated: 2026-07-01

Definition

The binomial theorem expands a power of a two-term sum into a sum of terms:

Each term pairs a binomial coefficient $\binom{n}{k} = \tfrac{n!}{k!\,(n-k)!}$ with $a$ raised to $(n-k)$ and $b$ raised to $k$. As $k$ runs from $0$ to $n$, the power of $a$ falls from $n$ to $0$ while the power of $b$ rises from $0$ to $n$, so the two exponents always sum to $n$.

General term

The single term at index $k$ (counting from $0$) is:

This is what the term-index field computes. Setting $k = 0$ gives the leading term $a^n$; setting $k = n$ gives the trailing term $b^n$. Reading one term at a time is handy when only a particular coefficient or the term carrying a given power is needed, without writing out the whole expansion.

Worked example

Take $(2 + 3)^4$ and read the term at $k = 2$. First the coefficient:

Then the full term $T_3 = \binom{4}{2}\, a^{\,n-k}\, b^{\,k}$:

The complete expansion is $(2 + 3)^4 = 16 + 96 + 216 + 216 + 81 = 625$, which matches $5^4$ as a check.

Coefficients and Pascal's triangle

The coefficients $\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}$ form row $n$ of Pascal's triangle, with the single-entry apex counted as row $0$. They obey the recurrence:

so each entry is the sum of the two directly above it. For $(a + b)^3$ the row is $1, 3, 3, 1$; for $(a + b)^4$ it is $1, 4, 6, 4, 1$. The coefficient chart above plots this row directly.

Symmetry

Binomial coefficients are symmetric:

so a row reads the same forwards and backwards. This mirrors the fact that choosing which $k$ factors contribute a $b$ is the same as choosing which $n-k$ factors contribute an $a$.

Sum of the coefficients

Setting $a = b = 1$ collapses the theorem to a compact identity:

The coefficients in any row therefore add up to a power of two — the row for $n = 3$ sums to $8$, the row for $n = 6$ sums to $64$. Setting $a = 1, b = -1$ gives the alternating sum, which is $0$ for every $n \geq 1$.

Common applications

• Algebra: expanding polynomials and simplifying expressions raised to a power.
• Probability: the coefficients drive the binomial distribution, where $\binom{n}{k} p^k (1-p)^{n-k}$ is the chance of $k$ successes in $n$ trials.
• Approximation: for small $x$, keeping the first two terms of $(1 + x)^n$ gives the estimate $1 + nx$.
• Combinatorics: identities such as the row sum $2^n$ and the alternating sum $0$ follow directly from the theorem.

Practical notes

Integer power required. The form above assumes $n$ is a whole number. Fractional or negative exponents use the generalized binomial series, an infinite sum that is a separate topic.

Leading and trailing terms. $\binom{n}{0} = \binom{n}{n} = 1$, so the first and last coefficients of every expansion are $1$.

For the coefficients on their own — the "$n$ choose $k$" values without the powers of $a$ and $b$ — use the Combination Calculator — C(n, r).