If \alpha+\beta+\gamma=\alpha\beta+\beta\gamma+\gamma\alpha, then what is (1-\alpha)(1-\beta)(1-\gamma) equal to?
- A. 1-\alpha\beta\gamma ✓
- B. 1+\alpha\beta\gamma
- C. \alpha^{2}+\beta^{2}+\gamma^{2}
- D. (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)
Correct Answer: A. 1-\alpha\beta\gamma
Explanation
Expanding the expression gives 1 - (\alpha+\beta+\gamma) + (\alpha\beta+\beta\gamma+\gamma\alpha) - \alpha\beta\gamma. Given \alpha+\beta+\gamma = \alpha\beta+\beta\gamma+\gamma\alpha, the two middle terms cancel out, leaving 1 - \alpha\beta\gamma.
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