Difference between revisions of "2019 AMC 8 Problems/Problem 17"
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==Solution 3== | ==Solution 3== | ||
Rewriting the numerator and the denominator, we get <math>\frac{\frac{100! \cdot 98!}{2}}{\left(99!\right)^2}</math>. We can simplify by canceling 99! on both sides, leaving us with: <math>\frac{100 \cdot 98!}{2 \cdot 99!}</math> We rewrite <math>99!</math> as <math>99 \cdot 98!</math> and cancel <math>98!</math>, which gets <math>\boxed{(B)\frac{50}{99}}</math>. | Rewriting the numerator and the denominator, we get <math>\frac{\frac{100! \cdot 98!}{2}}{\left(99!\right)^2}</math>. We can simplify by canceling 99! on both sides, leaving us with: <math>\frac{100 \cdot 98!}{2 \cdot 99!}</math> We rewrite <math>99!</math> as <math>99 \cdot 98!</math> and cancel <math>98!</math>, which gets <math>\boxed{(B)\frac{50}{99}}</math>. | ||
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+ | ==Solution 4== | ||
+ | Obviously all of the terms are less than one by difference of squares, so we eliminate options <math>(D)</math> and <math>(E)</math>. <math>(C)</math> is too close to one to be possible, and as stated in Solution 2 it gets close to but never actually becomes <math>1/2</math>. Therefore, by process of elimination we get that the answer is <math>B</math>. | ||
==Video Solution 1== | ==Video Solution 1== |
Revision as of 16:20, 20 December 2022
Contents
[hide]Problem
What is the value of the product
Solution 1 (Telescoping)
We rewrite:
The middle terms cancel, leaving us with
Solution 2
If you calculate the first few values of the equation, all of the values tend to close to , but are not equal to it. The answer closest to but not equal to it is .
Solution 3
Rewriting the numerator and the denominator, we get . We can simplify by canceling 99! on both sides, leaving us with: We rewrite as and cancel , which gets .
Solution 4
Obviously all of the terms are less than one by difference of squares, so we eliminate options and . is too close to one to be possible, and as stated in Solution 2 it gets close to but never actually becomes . Therefore, by process of elimination we get that the answer is .
Video Solution 1
https://www.youtube.com/watch?v=yPQmvyVyvaM
Associated video
https://www.youtube.com/watch?v=ffHl1dAjs7g&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=1
~ MathEx
Video Solution 2
Solution detailing how to solve the problem:
https://www.youtube.com/watch?v=VezsRMJvGPs&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=18
Video Solution 3
~savannahsolver
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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.