document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153346>Question 1153346</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.\r<BR>\n" );
document.write( "\n" );
document.write( "How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade?\r<BR>\n" );
document.write( "\n" );
document.write( "How many ways are there to sit the committee in a row at a meeting if the freshman, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #775578 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=775578\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Question:<BR>\n" );
document.write( "A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.<BR>\n" );
document.write( "How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=red>Answer: 1,244,160</font> (a little over 1 million)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------<BR>\n" );
document.write( "Work Shown:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The students must sit together by grade, so we effectively have 4 sections or blocks. There are 4! = 4*3*2*1 = 24 ways to arrange 4 blocks. Let A = 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Within the freshman block, there are 3! = 3*2*1 = 6 ways to arrange these students. Let B = 6.<BR>\n" );
document.write( "Within the sophomores block, there are 6! = 6*5*4*3*2*1 = 720 ways to arrange these students. Let C = 720.<BR>\n" );
document.write( "Within the juniors block, there are 3! = 3*2*1 = 6 ways to arrange these students. Let D = 6.<BR>\n" );
document.write( "Within the seniors block, there are 2! = 2*1 = 2 ways to arrange these students. Let E = 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Put together, there are A*B*C*D*E = 24*6*720*6*2 = <font color=red>1,244,160</font> different permutations possible such that the students must sit together by grade.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==============================================================<BR>\n" );
document.write( "Problem 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Question:<BR>\n" );
document.write( "A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.<BR>\n" );
document.write( "How many ways are there to sit the committee in a row at a meeting if the freshman, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=red>Answer: 3,110,400</font> (slightly over 3 million)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------<BR>\n" );
document.write( "Work Shown:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 3 blocks: freshmen, sophomores, juniors<BR>\n" );
document.write( "Then there are 2 seniors who can sit wherever they want. <BR>\n" );
document.write( "Effectively, there are 5 \"people\" (3 of which are groups of people)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A = 5! = 5*4*3*2*1 = 120 ways to arrange the five \"people\"<BR>\n" );
document.write( "B = 3! = 3*2*1 = 6 ways to arrange the freshmen within any particular block<BR>\n" );
document.write( "C = 6! = 6*5*4*3*2*1 = 720 ways to arrange the sophomores within any particular block<BR>\n" );
document.write( "D = 3! = 3*2*1 = 6 ways to arrange the juniors within any particular block<BR>\n" );
document.write( "A*B*C*D = 120*6*720*6 = <font color=red>3,110,400</font> is the total number of permutations in which the freshmen, sophomores, and juniors must sit together but the seniors can sit anywhere they want.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );