document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200164>Question 1200164</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could a tutor please verify whether the following solution is correct?<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1134057.html\r<BR>\n" );
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document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834206 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834206\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem: <BR>\n" );
document.write( "In a certain region all license plates are composed of three letters followed by three numbers, or three numbers followed by three letters. <BR>\n" );
document.write( "The only restriction is that zero can never be the first of the three numbers if the three numbers come first. <BR>\n" );
document.write( "Find how many license plates are possible.\r<BR>\n" );
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document.write( "We have two different options<ul><li>Option A: The three letters happen before the three numbers. Example: ABC123</li><li>Option B: The three numbers happen before the three letters. Zero cannot be listed first. Example: 123ABC</li></ul>In reference to this previous solution<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1134057.html<BR>\n" );
document.write( "The tutor @Glaviolette made an error in saying there are 35 options for the first slot.\r<BR>\n" );
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document.write( "If you go with option A, then the first slot must be a letter. There are 26 choices here. <BR>\n" );
document.write( "We won't have any numeric digits to choose from for this first slot.\r<BR>\n" );
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document.write( "If you go with option B, then the first slot is a nonzero number. <BR>\n" );
document.write( "There are 9 choices here (since we exclude zero). <BR>\n" );
document.write( "We won't have any letters to choose from for this first slot.\r<BR>\n" );
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document.write( "We cannot have an example plate of something like A23BCE because it mixes numbers with letters in each trio block.\r<BR>\n" );
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document.write( "Let's calculate the number of license plates if option A happens.<BR>\n" );
document.write( "In this case, we have 26^3 ways to write out the three letters assuming repeats are possible. <BR>\n" );
document.write( "Also, there are 10^3 ways to write out the three digits. The digit 0 can be first of the trio.<BR>\n" );
document.write( "There are 26^3*10^3 = 17,576,000 ways to have option A happen.\r<BR>\n" );
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document.write( "Now onto option B.<BR>\n" );
document.write( "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. <BR>\n" );
document.write( "That gives 9*10^2 different combos so far. <BR>\n" );
document.write( "Then we have 26^3 different ways to get three letters.<BR>\n" );
document.write( "There are 9*10^2*26^3 = 15,818,400 ways to have option B happen.\r<BR>\n" );
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document.write( "Options A and B are mutually exclusive. They also partition the sample space. <BR>\n" );
document.write( "Either one or the other event must happen, but not both simultaneously.<BR>\n" );
document.write( "These facts allow us to simply add the results we got in the previous paragraphs:<BR>\n" );
document.write( "A+B = (17,576,000) + (15,818,400)<BR>\n" );
document.write( "A+B = 33,394,400<BR>\n" );
document.write( "This is roughly 33.4 million different license plates.\r<BR>\n" );
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document.write( "Final Answer: <font color=red size=4>33,394,400</font><BR>\n" );
document.write( "Delete the commas if necessary.<BR>\n" );
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