What is the value of i(i-1)(i-2)(i-3), where i = √-1?

2025 AMC 12B Problem 3 — Solution

2025 AMC 12B Problem 3

Attempt Problem 3 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

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What is the value of $i(i-1)(i-2)(i-3),$ where $i = \sqrt{-1}?$

$6 - 5i$

$-10i$

$10i$

$-10$

$10$

Answer: D

Concepts:complex number

Difficulty rating: 1130

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.

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