Question 1200164: Could a tutor please verify whether the following solution is correct?
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1134057.html
Thanks!
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52769)   (Show Source):
You can put this solution on YOUR website! .
In a certain region all license plates are composed of three letters followed by three numbers,
or three numbers followed by three letters. The only restriction is that zero can never be
the first of the three numbers if the three numbers come first.
Find how many license plates are possible.
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From the first glance, I see that this solution in the referred source is overcomplicated.
I mean, the logic is overcomplicated.
Therefore, I developed my own simple (adequately simple) solution, in 3 clear steps.
(1) if 3 digits go first, followed by 3 letters, with the imposed restrictions,
then the number of plates is 9*10*10*26*26*26 = 15818400.
(2) if 3 letters go first, followed by 3 digits (no restrictions in this case)
then the number of plates is 26*26*26*10*10*10 = 17576000.
(3) The answer is the sum of these numbers 15818400 + 17576000 = 33394400.
Solved.
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Now I see that the referred solution is not only overcomplicated - it is INCORRECT, in addition.
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Formulation of the problem is not perfect.
To be accurate Math formulation, the term " number " should be replaced by " digit " everywhere.
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For the sake of next generations of students I submitted this my solution under the referred link.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Problem:
In a certain region all license plates are composed of three letters followed by three numbers, or three numbers followed by three letters.
The only restriction is that zero can never be the first of the three numbers if the three numbers come first.
Find how many license plates are possible.
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We have two different options- Option A: The three letters happen before the three numbers. Example: ABC123
- Option B: The three numbers happen before the three letters. Zero cannot be listed first. Example: 123ABC
In reference to this previous solution
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1134057.html
The tutor @Glaviolette made an error in saying there are 35 options for the first slot.
If you go with option A, then the first slot must be a letter. There are 26 choices here.
We won't have any numeric digits to choose from for this first slot.
If you go with option B, then the first slot is a nonzero number.
There are 9 choices here (since we exclude zero).
We won't have any letters to choose from for this first slot.
We cannot have an example plate of something like A23BCE because it mixes numbers with letters in each trio block.
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Let's calculate the number of license plates if option A happens.
In this case, we have 26^3 ways to write out the three letters assuming repeats are possible.
Also, there are 10^3 ways to write out the three digits. The digit 0 can be first of the trio.
There are 26^3*10^3 = 17,576,000 ways to have option A happen.
Now onto option B.
Since zero cannot be listed first, we have 9 choices for the first slot. Then 10 choices for each of the remaining two numeric slots.
That gives 9*10^2 different combos so far.
Then we have 26^3 different ways to get three letters.
There are 9*10^2*26^3 = 15,818,400 ways to have option B happen.
Options A and B are mutually exclusive. They also partition the sample space.
Either one or the other event must happen, but not both simultaneously.
These facts allow us to simply add the results we got in the previous paragraphs:
A+B = (17,576,000) + (15,818,400)
A+B = 33,394,400
This is roughly 33.4 million different license plates.
Final Answer: 33,394,400
Delete the commas if necessary.
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