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2024 AMC 12B Problem 2

Problem 2 of 25EasierCounting & Probability

What is 10!−7!⋅6!?10! - 7! \cdot 6!?

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Solution

Note that 6!=720=8⋅9⋅10.6! = 720 = 8 \cdot 9 \cdot 10. Therefore 7!⋅6!=7!⋅(8⋅9⋅10)=10!.7! \cdot 6! = 7! \cdot (8 \cdot 9 \cdot 10) = 10!. So 10!−7!⋅6!=10!−10!=0.10! - 7! \cdot 6! = 10! - 10! = 0. Thus, the correct 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.