document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=629274>Question 629274</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Different positive four-digit integers are to be formed by using each of the digits 1,2,3,4 just once in each integer. How many different such integers can be formed if the digits 3 and 4 must NEVER be next to each other?\r<BR>\n" );
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document.write( "A. 4<BR>\n" );
document.write( "B. 8<BR>\n" );
document.write( "C. 12<BR>\n" );
document.write( "D. 16<BR>\n" );
document.write( "E. 24 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #396147 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=396147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe the answer is going to be 12.<BR>\n" );
document.write( "here's how.<BR>\n" );
document.write( "assume that 3 and 4 HAVE to be together.<BR>\n" );
document.write( "They are treated as 1, and the formula becomes 3! rather than 4!.<BR>\n" );
document.write( "3! equals 6 ways.<BR>\n" );
document.write( "we'll let the number 7 represent 3 and 4 together.<BR>\n" );
document.write( "the 6 ways are:<BR>\n" );
document.write( "127<BR>\n" );
document.write( "172<BR>\n" );
document.write( "217<BR>\n" );
document.write( "271<BR>\n" );
document.write( "712<BR>\n" );
document.write( "721<BR>\n" );
document.write( "now we take the 7 and break it up into A and B so that they can be distinguished easier from the numbers 1 and 2.<BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "12AB<BR>\n" );
document.write( "12BA<BR>\n" );
document.write( "21AB<BR>\n" );
document.write( "21BA<BR>\n" );
document.write( "1AB2<BR>\n" );
document.write( "1BA2<BR>\n" );
document.write( "2AB1<BR>\n" );
document.write( "2BA1<BR>\n" );
document.write( "AB12<BR>\n" );
document.write( "AB21<BR>\n" );
document.write( "BA12<BR>\n" );
document.write( "BA21<BR>\n" );
document.write( "since 4 number can be arranged 4! ways with no restrictions, then the number of ways we can get the 4 numbers arranged when we can't have 3 and 4 together would have to be 24 - 12 = 12.<BR>\n" );
document.write( "those ways would be:<BR>\n" );
document.write( "1A2B<BR>\n" );
document.write( "1B2A<BR>\n" );
document.write( "2A1B<BR>\n" );
document.write( "2B1A<BR>\n" );
document.write( "A1B2<BR>\n" );
document.write( "A2B1<BR>\n" );
document.write( "B1A2<BR>\n" );
document.write( "B2A1<BR>\n" );
document.write( "A12B<BR>\n" );
document.write( "B12A<BR>\n" );
document.write( "A21B<BR>\n" );
document.write( "A12B<BR>\n" );
document.write( "just replace the A and B with 3 and 4 and you have your answer in the format the problem was presented.<BR>\n" );
document.write( "this works with 3 numbers as well.<BR>\n" );
document.write( "suppose your number were 1 and 2 and 3.<BR>\n" );
document.write( "supposed you wanted to know how many ways you could get the numbers when the 2 and the 3 could not be together.<BR>\n" );
document.write( "assume they had to be together.<BR>\n" );
document.write( "you then get 2! ways that can happen.<BR>\n" );
document.write( "you then multiply that by 2 to get 4 ways that can happen.<BR>\n" );
document.write( "assign A and B to 2 and 3 and you get:<BR>\n" );
document.write( "1AB<BR>\n" );
document.write( "1BA<BR>\n" );
document.write( "AB1<BR>\n" );
document.write( "BA1<BR>\n" );
document.write( "that's the 4 ways they can be together.<BR>\n" );
document.write( "3! = 6<BR>\n" );
document.write( "6 - 4 = 2<BR>\n" );
document.write( "that mans 2 ways that cannot be together.<BR>\n" );
document.write( "those ways are:<BR>\n" );
document.write( "A1B<BR>\n" );
document.write( "B1A<BR>\n" );
document.write( "replace A with 2 and B with 3 and you get the answer in the format it was presented.<BR>\n" );
document.write( "I believe you answer is 12.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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