Difference between revisions of "2024 AMC 10B Problems/Problem 2"

(Solution 1)
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Factoring out <math>7!</math> gives <cmath>7!(10\cdot9\cdot8-1\cdot6!).</cmath> Since <math>10\cdot9\cdot8=6!=720</math>, the answer is <math>\boxed{\text{(B) }0}</math> ~Tacos_are_yummy_1
 
Factoring out <math>7!</math> gives <cmath>7!(10\cdot9\cdot8-1\cdot6!).</cmath> Since <math>10\cdot9\cdot8=6!=720</math>, the answer is <math>\boxed{\text{(B) }0}</math> ~Tacos_are_yummy_1
  
[b][u]Remark[/b][/u]
 
 
Factoring <math>6!</math> also works, it just makes the expression in the parenthesis a little harder to compute.
 
Factoring <math>6!</math> also works, it just makes the expression in the parenthesis a little harder to compute.
  

Revision as of 11:44, 14 November 2024

The following problem is from both the 2024 AMC 10B #2 and 2024 AMC 12B #2, so both problems redirect to this page.

Problem

What is $10! - 7! \cdot 6!$

$\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720$


Certain China testpapers:

What is $10! - 7! \cdot 6! - 5!$

$\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720$

Solution 1

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

$6! \cdot 7! = 720 \cdot 7!$

Therefore, the equation is equal to $720 \cdot 7! - 720 \cdot 7! = \boxed{\textbf{(B) }0}$

Solution for certain China test papers:

$0 - 5! = \boxed{\textbf{(A) }-120}$

~Aray10 (Main Solution) and RULE101 (Modifications for certain China test papers)

Solution 1

Factoring out $7!$ gives \[7!(10\cdot9\cdot8-1\cdot6!).\] Since $10\cdot9\cdot8=6!=720$, the answer is $\boxed{\text{(B) }0}$ ~Tacos_are_yummy_1

Factoring $6!$ also works, it just makes the expression in the parenthesis a little harder to compute.

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/DIl3rLQQkQQ?feature=shared

~ Pi Academy

See also

2024 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2024 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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