Difference between revisions of "2020 AMC 8 Problems/Problem 12"
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| Line 8: | Line 8: | ||
<cmath>10!*12=12*N!.</cmath> | <cmath>10!*12=12*N!.</cmath> | ||
| − | We can cancel the <math> | + | We can cancel the <math>12</math>'s, since we are multiplying them on both sides of the equation. |
We have | We have | ||
<cmath>10!=N!.</cmath> | <cmath>10!=N!.</cmath> | ||
| − | From here, it is obvious that <math>N=10(A).</math> | + | From here, it is obvious that <math>N=\boxed{10\textbf{(A)}}.</math> |
-iiRishabii | -iiRishabii | ||
Revision as of 23:59, 17 November 2020
For positive integers 
, the notation 
denotes 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
Notice that 
= 
and we can combine the numbers to create a larger factorial. To turn 
into 
we need to multiply by 
which equals to 
Therefore, we have
![\[10!*12=12*N!.\]](http://latex.artofproblemsolving.com/b/5/7/b57d51cc68f940f63a1dde3a63bbc125429d2bfc.png)
We can cancel the 
's, since we are multiplying them on both sides of the equation.
We have
![\[10!=N!.\]](http://latex.artofproblemsolving.com/d/d/8/dd8b5a203f6c403ffb8a87279e697ee510eefb36.png)
From here, it is obvious that 
-iiRishabii
