Difference between revisions of "2020 AMC 8 Problems/Problem 12"
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==Problem== | ==Problem== | ||
| − | For a positive integer <math>n</math>, the factorial notation <math>n!</math> represents the product of the integers from <math>n</math> to <math>1</math>. | + | For a positive integer <math>n</math>, the factorial notation <math>n!</math> represents the product of the integers from <math>n</math> to <math>1</math>. What value of <math>N</math> satisfies the following equation? <cmath>5!\cdot 9!=12\cdot N!</cmath> |
<math>\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14\qquad</math> | <math>\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14\qquad</math> | ||
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==Solution 3 (using answer choices)== | ==Solution 3 (using answer choices)== | ||
| − | We can see that the answers <math>\textbf{(B)}</math> to <math>\textbf{(E)}</math> contain a factor of <math>11</math>, but there is no such factor of <math>11</math> in <math>5! \cdot 9!</math>. Therefore, the | + | We can see that the answers <math>\textbf{(B)}</math> to <math>\textbf{(E)}</math> contain a factor of <math>11</math>, but there is no such factor of <math>11</math> in <math>5! \cdot 9!</math>. Therefore, the answer must be <math>\boxed{\textbf{(A) }10}</math>. |
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/9k59v-Fr3aE | ||
| + | |||
| + | ~savannahsolver | ||
==Video Solution== | ==Video Solution== | ||
| − | https://youtu.be/ | + | https://youtu.be/xjwDsaRE_Wo |
| + | |||
| + | ==Video Solution by Interstigation== | ||
| + | https://youtu.be/YnwkBZTv5Fw?t=504 | ||
| + | |||
| + | ~Interstigation | ||
==See also== | ==See also== | ||
{{AMC8 box|year=2020|num-b=11|num-a=13}} | {{AMC8 box|year=2020|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 14:32, 18 April 2021
Contents
Problem
For a positive integer 
, the factorial notation 
represents the product of the integers from to 
. What value of 
satisfies the following equation? ![\[5!\cdot 9!=12\cdot N!\]](http://latex.artofproblemsolving.com/7/a/a/7aaf22ceff79c5a5063175866cf2cba2e4a57f9b.png)
Solution 1
We have 
, and 
. Therefore the equation becomes 
, and so 
. Cancelling the 
s, it is clear that 
.
Solution 2 (variant of Solution 1)
Since 
, we obtain 
, which becomes 
and thus 
. We therefore deduce .
Solution 3 (using answer choices)
We can see that the answers 
to 
contain a factor of 
, but there is no such factor of in 
. Therefore, the answer must be 
.
Video Solution by WhyMath
~savannahsolver
Video Solution
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=504
~Interstigation
See also
| 2020 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
