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2024 AMC 10B Problem 2

Problem 2 of 25EasierCounting & Probability

What is 10!−7!⋅6!?10! - 7! \cdot 6!?

Answer choices

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Solution

Write 10!=10⋅9⋅8⋅7!=720⋅7!.10! = 10 \cdot 9 \cdot 8 \cdot 7! = 720 \cdot 7!. But 720=6!720 = 6! too, so 10!=6!⋅7!=7!⋅6!.10! = 6! \cdot 7! = 7! \cdot 6!. The two terms are the same. That makes 10!−7!⋅6!=0.10! - 7! \cdot 6! = 0. Therefore, the answer is B.

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Concepts: factorial

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