Difference between revisions of "2016 AMC 10A Problems/Problem 1"

Latest revision as of 21:59, 1 November 2023

Problem

What is the value of $\dfrac{11!-10!}{9!}$ ?

$\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

Solution 1

We can use subtraction of fractions to get $\frac{11!-10!}{9!} = \frac{11!}{9!} - \frac{10!}{9!} = 110 -10 = \boxed{\textbf{(B)}\;100}.$

Solution 2

Factoring out $9!$ gives $\frac{11!-10!}{9!} = \frac{9!(11 \cdot 10 - 10)}{9!} = 110-10=\boxed{\textbf{(B)}~100}$ .

Solution 3

$\dfrac{11!-10!}{9!}$ consider 10 as n $\dfrac{(n+1)!-n!}{(n-1)!}$ simpify $\dfrac{(n+1)n!-(-1)n!}{(n-1)!}$ = $\dfrac{n(n!)}{(n-1)!}$ = $\dfrac{n(n(n-1)!)}{(n-1)!}$ = $\dfrac{n(n)(1)}{(1}$ = $\dfrac{n^2}{1}$ subsitute n as 10 again $\dfrac{10^2}{1}$

answer is $10^2$ which is 100

Solution 4

We are given the equation $\frac{11!-10!}{9!}$

This is equivalent to $\frac{11(10!) - 1(10!)}{9!}$ Simplifying, we get $\frac{(11-1)(10!)}{9!}$ , which equals $10 \cdot 10$ .

Therefore, the answer is $10^2$ = $\boxed{\textbf{(B)}~100}$ .

~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

2016 AMC 10A (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions

2016 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 7: / Line 7: @@
 ==Solution 1==
-Factoring out the <math>10!</math> from the numerator and cancelling out the <math>9!</math>s in the numerator and denominator, we have: <cmath>\frac{11!-10!}{9!} = \frac{11 \cdot 10! - 1 \cdot 10!}{9!} =  \frac{(11 - 1) \cdot (10!)}{9!} = 10 \cdot 10 =\boxed{\textbf{(B)}\;100.}</cmath>
+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 2==
-We can use subtraction of fractions to get <cmath>\frac{11!-10!}{9!} = \frac{11!}{9!} - \frac{10!}{9!}</cmath> which will get us <math>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
-Factoring out <math>9!</math>
 ==See Also==

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