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

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== Problem ==
 
== 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?  
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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>
  
<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.  
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 \mathrm{C}</math>.
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== Video Solutions ==
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https://youtu.be/3MiGotKnC_U?t=1446
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~ThePuzzlr
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https://youtu.be/0W3VmFp55cM?t=847
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~ pi_is_3.14
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https://youtu.be/f56aG55oG8w
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~savannahsolver
  
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}}
  
== See Also ==
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[[Category:Introductory Combinatorics Problems]]
*[[2006 AMC 10A Problems]]
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{{MAA Notice}}

Latest revision as of 09:08, 19 December 2021

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 $5\times 10^4\times 26^2 \Longrightarrow \mathrm{C}$.

Video Solutions

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

~ThePuzzlr

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

~ pi_is_3.14

https://youtu.be/f56aG55oG8w

~savannahsolver

See also

2006 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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All AMC 10 Problems and Solutions

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