Binomial theorem
Multi-binomial theorem

When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to (x 1 + y 1) n 1 ⋯ (x d + y d) n d = ∑ k 1 = 0 n 1 ⋯ ∑ k d = 0 n d (n 1 k 1) x 1 k 1 y 1 n 1 − k 1 … (n d k d) x d k d y d n d − k d. {\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.}

This may be written more concisely, by multi-index notation, as (x + y) α = ∑ ν ≤ α (α ν) x ν y α − ν. {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.}
oulipo:avoid_letter[a], combination:repeat_prompt, punctuation:no_comma