Difference between revisions of "2016 AMC 10A Problems/Problem 1"
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| + | ==Problem== | ||
| + | |||
What is the value of <math>\dfrac{11!-10!}{9!}</math>? | What is the value of <math>\dfrac{11!-10!}{9!}</math>? | ||
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math> | <math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math> | ||
| + | |||
| + | ==Solution 1== | ||
| + | |||
| + | We can use subtraction of fractions to get <cmath>\frac{11!-10!}{9!} = \frac{11!}{9!} - \frac{10!}{9!} = 110 -10 = \boxed{\textbf{(B)}\;100}.</cmath> | ||
| + | |||
| + | ==Solution 2== | ||
| + | |||
| + | Factoring out <math>9!</math> gives <math>\frac{11!-10!}{9!} = \frac{9!(11 \cdot 10 - 10)}{9!} = 110-10=\boxed{\textbf{(B)}~100}</math>. | ||
| + | |||
| + | |||
| + | ==Solution 3== | ||
| + | <math>\dfrac{11!-10!}{9!}</math> | ||
| + | consider 10 as n | ||
| + | <math>\dfrac{(n+1)!-n!}{(n-1)!}</math> | ||
| + | simpify | ||
| + | <math>\dfrac{(n+1)n!-(-1)n!}{(n-1)!}</math> = <math>\dfrac{n(n!)}{(n-1)!}</math> = <math>\dfrac{n(n(n-1)!)}{(n-1)!}</math> = <math>\dfrac{n(n)(1)}{(1}</math> = <math>\dfrac{n^2}{1}</math> | ||
| + | subsitute n as 10 again | ||
| + | <math>\dfrac{10^2}{1}</math> | ||
| + | |||
| + | answer is <math>10^2</math> which is 100 | ||
| + | |||
| + | ==Solution 4== | ||
| + | We are given the equation <math>\frac{11!-10!}{9!}</math> | ||
| + | |||
| + | This is equivalent to <math>\frac{11(10!) - 1(10!)}{9!}</math> | ||
| + | Simplifying, we get <math>\frac{(11-1)(10!)}{9!}</math>, which equals <math>10 \cdot 10</math>. | ||
| + | |||
| + | Therefore, the answer is <math>10^2</math> = <math>\boxed{\textbf{(B)}~100}</math>. | ||
| + | |||
| + | ~TheGoldenRetriever | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
| + | https://youtu.be/r5G98oPPyNM | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | |||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/VIt6LnkV4_w | ||
| + | |||
| + | |||
| + | https://youtu.be/CrS7oHDrvP8 | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution (FASTEST METHOD!)== | ||
| + | |||
| + | https://youtu.be/jowREGsZaTs | ||
| + | |||
| + | ~Veer Mahajan | ||
| + | |||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2016|ab=A|before=First Problem|num-a=2}} | ||
| + | {{AMC12 box|year=2016|ab=A|before=First Problem|num-a=2}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 22:59, 1 November 2023
Contents
Problem
What is the value of 
?
Solution 1
We can use subtraction of fractions to get ![\[\frac{11!-10!}{9!} = \frac{11!}{9!} - \frac{10!}{9!} = 110 -10 = \boxed{\textbf{(B)}\;100}.\]](http://latex.artofproblemsolving.com/5/d/8/5d8e4e6c76063dc4767a679ab11d828b652b9470.png)
Solution 2
Factoring out 
gives 
.
Solution 3
consider 10 as n
simpify
= 
= 
= 
= 
subsitute n as 10 again
answer is 
which is 100
Solution 4
We are given the equation 
This is equivalent to 
Simplifying, we get 
, which equals 
.
Therefore, the answer is = 
.
~TheGoldenRetriever
Video Solution (HOW TO THINK CREATIVELY!!!)
https://youtu.be/r5G98oPPyNM
~Education, the Study of Everything
Video Solution
~savannahsolver
Video Solution (FASTEST METHOD!)
~Veer Mahajan
See Also
| 2016 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 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 | ||
| 2016 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by First Problem |
Followed by Problem 2 |
| 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 | |
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