2024 AMC 10B Problems/Problem 2: Difference between revisions

Latest revision as of 21:10, 6 October 2025

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 $10! - 7! \cdot 6!$

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

[ONLY FOR CERTAIN CHINESE 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 $720 \cdot 7! - 720 \cdot 7! = \boxed{\textbf{(B) }0}$

[ONLY FOR CERTAIN CHINESE TESTPAPERS]

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

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

Solution 2

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

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

Solution 3

Note that $10! - 7! \cdot 6!$ must be divisible by $7$ , and $\boxed{\text{(B) }0}$ is the only option divisible by $7$ .

Solution 4

$10! - 7! \cdot 6!$ can be split into two parts, $10!$ and $7! \cdot 6!$ . We can break the $6!$ into $(2 \cdot 4)(3 \cdot 5 \cdot 6)$ The $2 \cdot 4$ part makes $8$ , and the $3 \cdot 5 \cdot 6$ part makes $90$ , which is $9 \cdot 10$ . We still have the 7!, and we can multiply it by $8 \cdot 9 \cdot 10$ . This is clearly equivalent to $10!$ , so our solution is $10! - 10! =$ $\boxed{\text{(B) }0}$ .

Solution 5

$10! = 3,628,800$ , $7! = 5,040$ , and $6! = 720$ . Of course, if you're fast enough, you can do $5,040 \cdot 720 = 3,628,800$ . Therefore, $3,628,800 - 3,628,800 = \boxed{\text{(B) }0}$ .

-pepper2831

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

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

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

Video Solution by Daily Dose of Math

https://youtu.be/DVlOz24jWuo

~Thesmartgreekmathdude

Video Solution by the BeautyofMath

https://youtu.be/mpnm_r6D3-w?t=373

~IceMatrix

2024 AMC 10B (Problems • Answer Key • Resources)
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 (Problems • Answer Key • Resources)
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 1: / Line 1: @@
-How many positive perfect humans less than 2024 hours old evenly divides into early access to the AMC 10B Problem 2 2024?
+{{duplicate|[[2024 AMC 10B Problems/Problem 2|2024 AMC 10B #2]] and [[2024 AMC 12B Problems/Problem 2|2024 AMC 12B #2]]}}
-A: -5% B: 0% C: 0.0000.-1% D: pi ^ -1 % E: |Nothing|
+==Problem==
+What is <math>10! - 7! \cdot 6!</math>
-(Personally, I'd go for E, no numbers to deal with, eh?)
+<math>\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720</math>
+[ONLY FOR CERTAIN CHINESE TESTPAPERS]
+What is <math>10! - 7! \cdot 6! - 5!</math>
+<math>\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720</math>
+==Solution 1==
+<math>10! = 10 \cdot 9 \cdot 8 \cdot 7! = 720 \cdot 7!</math>
+<math>6! \cdot 7! = 720 \cdot 7!</math>
+Therefore, the equation is equal to <math>720 \cdot 7! - 720 \cdot 7! = \boxed{\textbf{(B) }0}</math>
+[ONLY FOR CERTAIN CHINESE TESTPAPERS]
+<math>0 - 5! = \boxed{\textbf{(A) }-120}</math>
+~Aray10 (Main Solution) and RULE101 (Modifications for certain China test papers)
+==Solution 2==
+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>
+Factoring <math>6!</math> also works, it just makes the expression in the parenthesis a little harder to compute.
+==Solution 3==
+Note that <math>10! - 7! \cdot 6!</math> must be divisible by <math>7</math>, and <math>\boxed{\text{(B) }0}</math> is the only option divisible by <math>7</math>.
+==Solution 4==
+<math>10! - 7! \cdot 6!</math> can be split into two parts, <math>10!</math> and <math>7! \cdot 6!</math>.
+We can break the <math>6!</math> into <math>(2 \cdot 4)(3 \cdot 5 \cdot 6)</math>
+The <math>2 \cdot 4</math> part makes <math>8</math>, and the <math>3 \cdot 5 \cdot 6</math> part makes <math>90</math>, which is <math>9 \cdot 10</math>.
+We still have the 7!, and we can multiply it by <math>8 \cdot 9 \cdot 10</math>. This is clearly equivalent to <math>10!</math>, so our solution is <math>10! - 10! = </math><math>\boxed{\text{(B) }0}</math>.
+==Solution 5==
+<math>10! = 3,628,800</math>, <math>7! = 5,040</math>, and <math>6! = 720</math>. Of course, if you're fast enough, you can do <math>5,040 \cdot 720 = 3,628,800</math>. Therefore, <math>3,628,800 - 3,628,800 = \boxed{\text{(B) }0}</math>.
+-pepper2831
+==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)==
+https://youtu.be/DIl3rLQQkQQ?feature=shared
+~ Pi Academy
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=24EZaeAThuE
+== Video Solution by Daily Dose of Math ==
+https://youtu.be/DVlOz24jWuo
+~Thesmartgreekmathdude
+==Video Solution by the BeautyofMath==
+https://youtu.be/mpnm_r6D3-w?t=373
+~IceMatrix
+==See also==
+{{AMC10 box|year=2024|ab=B|num-b=1|num-a=3}}
+{{AMC12 box|year=2024|ab=B|num-b=1|num-a=3}}
+{{MAA Notice}}

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