2025 AMC 12B Problem 3

Problem 3 of 25EasierAlgebra

What is the value of

i(i-1)(i-2)(i-3),

where

i = \sqrt{-1}?

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Solution

i(i-1) = i^2 - i = -1 - i,

and

(i-2)(i-3) = i^2 - 5i + 6

= 5 - 5i.

Then

\begin{gathered} (-1-i)(5-5i) = -5 \\ {}+ 5i - 5i \\ {}+ 5i^2 \\ = -5 - 5 = -10. \end{gathered}

Thus, the correct answer is D.

Concepts: complex number

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.