document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175991>Question 1175991</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 5 boys and 4 girls forming a line to pay for the ticket to watch Frozen 2. In how many ways can they be arranged if 3 particular persons must stand beside each other? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #801818 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=801818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's say these 5 boys and 4 girls are students of a small classroom. Let's also say that the teacher is chaperoning them.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The 5+4 = 9 students are standing in line, and let's say that 3 adjacent students (standing next to each other) leave the line to get popcorn. The teacher can step in for those 3 students as a substitution. Wherever the teacher is, the three students will replace him/her when the students get back from buying popcorn.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the 9 people in line drops to 9-3 = 6 people after those 3 students leave. But then the teacher steps in to bump the count to 6+1 = 7 people.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those 7 people arrange in 7! = 7*6*5*4*3*2*1 = 5040 ways. <BR>\n" );
document.write( "The exclamation mark indicates factorial. We count down 7,6,5,... all the way to 1 multiplying along the way.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 5040 ways to arrange the group of students+teacher. Order matters. Of any of those 5040 permutations, there are 3! = 3*2*1 = 6 ways to arrange the trio of students who left the line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consider those three students to have codenames of A,B,C<BR>\n" );
document.write( "Those 6 arrangements are such<BR>\n" );
document.write( "A,B,C<BR>\n" );
document.write( "A,C,B<BR>\n" );
document.write( "B,A,C<BR>\n" );
document.write( "B,C,A<BR>\n" );
document.write( "C,A,B<BR>\n" );
document.write( "C,B,A\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that we'll have 6*5040 = <font color=red>30,240 different ways</font> to form the line such that those 3 students must stand next to one another. At this point, the teacher has been removed from the line.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );