The Binomial Theorem Sarmate.net --- conversion example 17 mai 2026 Binomial coefficients For non-negative integers $n$ and $k$ with $k \leq n$, the binomial coefficient is the number \[ \binom{n}{k} = \frac{n!}{k!\,(n-k)!}. \]

It counts the number of $k$-element subsets of an $n$-element set.

For any real numbers $a$ and $b$ and any positive integer $n$, $$ \begin{align} (a + b)^n & = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \tag{1} \\ & = a^{n} + n\,a^{n-1}b + \cdots + n\,a\,b^{n-1} + b^{n}. \tag{2} \end{align} $$

This identity is known as the binomial theorem.

Pascal's rule

The binomial coefficients satisfy the identity

\[ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}, \]

which generates Pascal's triangle row by row.

 Example Expand $(x + 2)^4$.

Using the binomial theorem with $a = x$ and $b = 2$:

\[ (x + 2)^4 = x^4 + 4 \cdot x^3 \cdot 2 + 6 \cdot x^2 \cdot 4 + 4 \cdot x \cdot 8 + 16, \]

which simplifies to

\[ (x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16. \] About

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