What is 10! - 7! × 6!?

2024 AMC 10B Problem 2

Attempt Problem 2 of the 2024 AMC 10B 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 2024 AMC 10B solutions, or check the answer key.

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What is

10! - 7! \cdot 6!?

-120

0

120

600

720

Write

10! = 10 \cdot 9 \cdot 8 \cdot 7!

Notice that

10 \cdot 9 \cdot 8

equals

6!

Write

10! = 10 \cdot 9 \cdot 8 \cdot 7! = 720 \cdot 7!.

But

720 = 6!

too, so

10! = 6! \cdot 7! = 7! \cdot 6!.

The two terms are the same. That makes

10! - 7! \cdot 6! = 0.

Therefore, the answer is B.