Difference between revisions of "2020 AMC 8 Problems/Problem 12"
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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> | 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> | ||
==Solution 1== | ==Solution 1== |
Revision as of 17:32, 18 November 2020
For a positive integer , the factorial notation
represents the product of the integers from
to
. What value of
satisfies the following equation?
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
We can cancel the
's, since we are multiplying them on both sides of the equation.
We have
From here, it is obvious that
-iiRishabii
Solution 2
.
~ junaidmansuri
Solution 3 (Non-rigorous)
We can see that the answers B through E have the factor 11, but there is no 11 in . Therefore, the answer must be the only answer without a
factor,
.
~Windigo
Solution 4
Notice that . We are also told that
from where it is obvious that
.
-franzliszt
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 |
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