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Toolkit 26

Expansion (or factorization) of three binomials

abc−ab−ac−bc+a+b+c−1=(a−1)(b−1)(c−1)abc-ab-ac-bc+a+b+c-1=(a-1)(b-1)(c-1)

Proof

First,

(b−1)(c−1)=bc−b−c+1.(b-1)(c-1)=bc-b-c+1.

Therefore,

(a−1)(b−1)(c−1)=(a−1)(bc−b−c+1)=abc−ab−ac+a−bc+b+c−1=abc−ab−ac−bc+a+b+c−1.□\begin{aligned} (a-1)(b-1)(c-1) &= (a-1)(bc-b-c+1) \\ &= abc-ab-ac+a-bc+b+c-1 \\ &= abc-ab-ac-bc+a+b+c-1. \quad\square \end{aligned}