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Difference between revisions of "2006 AMC 10A Problems/Problem 18"

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== Problem ==
 
== Problem ==
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A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?  
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A license plate in a certain state consists of <math>4</math> digits, not necessarily distinct, and <math>2</math> letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?  
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<math>\textbf{(A) } 10^4\times 26^2\qquad\textbf{(B) } 10^3\times 26^3\qquad\textbf{(C) } 5\times 10^4\times 26^2\qquad\textbf{(D) } 10^2\times 26^4\qquad\textbf{(E) } 5\times 10^3\times 26^3\qquad</math>
  
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<math>\mathrm{(A) \ } 10^4\times 26^2\qquad\mathrm{(B) \ } 10^3\times 26^3\qquad\mathrm{(C) \ } 5\times 10^4\times 26^2\qquad\mathrm{(D) \ } 10^2\times 26^4\qquad\mathrm{(E) \ } 5\times 10^3\times 26^3\qquad</math>
 
 
== Solution ==
 
== Solution ==
 
There are <math>10\cdot10\cdot10\cdot10 = 10^4</math> ways to choose 4 digits.
 
There are <math>10\cdot10\cdot10\cdot10 = 10^4</math> ways to choose 4 digits.
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For the letters to be next to each other, they can be the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters.  
 
For the letters to be next to each other, they can be the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters.  
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So, there are <math>5</math> choices for position.
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So, there are <math>6 - 1 = 5</math> choices for the position of the letters.
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Therefore, the number of distinct license plates is  <math> 5\times 10^4\times 26^2 \Longrightarrow \boxed{\mathrm{C}}</math>.
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== Solution 2 (Answer choices) ==
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There are <math>10^4</math> ways to choose the digits and <math>26^2</math> ways to choose the 2 letters. However, since the letters are next to each other, the result must be <math>10^4\cdot 26^2 \cdot \text{something}</math>. The only answer choice that matches this is <math>\boxed{\mathrm{C}}</math>.
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~ Andrew2019
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==Video Solution by OmegaLearn==
 +
https://youtu.be/0W3VmFp55cM?t=847
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== Video Solutions ==
 +
https://youtu.be/3MiGotKnC_U?t=1446
 +
 
 +
~ThePuzzlr
 +
 
 +
 
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https://youtu.be/f56aG55oG8w
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~savannahsolver
  
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Therefore there are  <math> 5\times 10^4\times 26^2 </math> distinct license plates <math> \Rightarrow C </math>
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== See also ==
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{{AMC10 box|year=2006|ab=A|num-b=17|num-a=19}}
  
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== See Also ==
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[[Category:Introductory Combinatorics Problems]]
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*[[2006 AMC 10A Problems]]
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{{MAA Notice}}

Latest revision as of 14:33, 30 July 2024

Problem

A license plate in a certain state consists of $4$ digits, not necessarily distinct, and $2$ letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?

$\textbf{(A) } 10^4\times 26^2\qquad\textbf{(B) } 10^3\times 26^3\qquad\textbf{(C) } 5\times 10^4\times 26^2\qquad\textbf{(D) } 10^2\times 26^4\qquad\textbf{(E) } 5\times 10^3\times 26^3\qquad$

Solution

There are $10\cdot10\cdot10\cdot10 = 10^4$ ways to choose 4 digits.

There are $26 \cdot 26 = 26^2$ ways to choose the 2 letters.

For the letters to be next to each other, they can be the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters. So, there are $6 - 1 = 5$ choices for the position of the letters.

Therefore, the number of distinct license plates is $5\times 10^4\times 26^2 \Longrightarrow \boxed{\mathrm{C}}$.

Solution 2 (Answer choices)

There are $10^4$ ways to choose the digits and $26^2$ ways to choose the 2 letters. However, since the letters are next to each other, the result must be $10^4\cdot 26^2 \cdot \text{something}$. The only answer choice that matches this is $\boxed{\mathrm{C}}$.

~ Andrew2019

Video Solution by OmegaLearn

https://youtu.be/0W3VmFp55cM?t=847

Video Solutions

https://youtu.be/3MiGotKnC_U?t=1446

~ThePuzzlr


https://youtu.be/f56aG55oG8w

~savannahsolver

See also

2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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

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