Difference between revisions of "2015 AMC 8 Problems/Problem 11"

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==Solution 1==
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===Solution 1===
 
There is one favorable case, which is the license plate says "AMC8".  We must now find how many total cases there are.  There are <math>5</math> choices for the first letter (since it must be a vowel), <math>21</math> choices for the second letter (since it must be of <math>21</math> consonants), <math>20</math> choices for the third letter (since it must differ from the second letter), and <math>10</math> choices for the number.  This leads to <math>5 \cdot 21 \cdot 20 \cdot 10=21000</math> total possible license plates.  That means the probability of a license plate saying "AMC8" is <math>\boxed{\textbf{(B) } \frac{1}{21,000}}</math>.
 
There is one favorable case, which is the license plate says "AMC8".  We must now find how many total cases there are.  There are <math>5</math> choices for the first letter (since it must be a vowel), <math>21</math> choices for the second letter (since it must be of <math>21</math> consonants), <math>20</math> choices for the third letter (since it must differ from the second letter), and <math>10</math> choices for the number.  This leads to <math>5 \cdot 21 \cdot 20 \cdot 10=21000</math> total possible license plates.  That means the probability of a license plate saying "AMC8" is <math>\boxed{\textbf{(B) } \frac{1}{21,000}}</math>.
  

Revision as of 23:26, 6 December 2015

In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?

$\textbf{(A) } \frac{1}{22,050} \qquad \textbf{(B) } \frac{1}{21,000}\qquad \textbf{(C) } \frac{1}{10,500}\qquad \textbf{(D) } \frac{1}{2,100} \qquad \textbf{(E) } \frac{1}{1,050}$

Solution 1

There is one favorable case, which is the license plate says "AMC8". We must now find how many total cases there are. There are $5$ choices for the first letter (since it must be a vowel), $21$ choices for the second letter (since it must be of $21$ consonants), $20$ choices for the third letter (since it must differ from the second letter), and $10$ choices for the number. This leads to $5 \cdot 21 \cdot 20 \cdot 10=21000$ total possible license plates. That means the probability of a license plate saying "AMC8" is $\boxed{\textbf{(B) } \frac{1}{21,000}}$.

Solution 2

The probability of choosing A as the first letter is $\dfrac{1}{5}$. The probability of choosing $M$ next is $\dfrac{1}{21}$. The probability of choosing C as the third letter is $\dfrac{1}{20}$ (since there are $20$ other consonants to choose from other then M). The probability of having $8$ as the last number is $\dfrac{1}{10}$. We multiply all these to obtain $\dfrac{1}{5}\cdot \dfrac{1}{21} \cdot \dfrac{1}{20}\cdot \dfrac{1}{10}=\dfrac{1}{5\times 21\times 20\times 10}=\dfrac{1}{21\times 100\times 10}=\boxed{\textbf{(B)}~\dfrac{1}{21,000}}$

See Also

2015 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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

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