document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191617>Question 1191617</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A dance instructor asks each student to do 4 out of the 10 dance routines. Of the 10 dance routines, 2 are easy, 5 are moderately difficult, and 3 are difficult. In how many ways can a student select each of the following for the 4 dance routines?<BR>\n" );
document.write( " a) 4 moderately difficult routines<BR>\n" );
document.write( " b) 4 easy or moderately difficult routines<BR>\n" );
document.write( " c) 2 moderately difficult and 2 difficult routines<BR>\n" );
document.write( " d) 1 easy and 3 difficult routines\r<BR>\n" );
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document.write( "(I want detailed explanation please and thank you). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849110 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849110\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this combination problem:\r<BR>\n" );
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document.write( "**a) 4 moderately difficult routines:**\r<BR>\n" );
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document.write( "* There are 5 moderately difficult routines.<BR>\n" );
document.write( "* The student must choose 4 of them.<BR>\n" );
document.write( "* Number of ways = ⁵C₄ = 5! / (4! * 1!) = 5\r<BR>\n" );
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document.write( "**b) 4 easy or moderately difficult routines:**\r<BR>\n" );
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document.write( "* There are 2 easy + 5 moderately difficult = 7 routines that are either easy or moderately difficult.<BR>\n" );
document.write( "* The student must choose 4 of them.<BR>\n" );
document.write( "* Number of ways = ⁷C₄ = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35\r<BR>\n" );
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document.write( "**c) 2 moderately difficult and 2 difficult routines:**\r<BR>\n" );
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document.write( "* There are 5 moderately difficult routines, and the student must choose 2: ⁵C₂ = 5! / (2! * 3!) = 10<BR>\n" );
document.write( "* There are 3 difficult routines, and the student must choose 2: ³C₂ = 3! / (2! * 1!) = 3<BR>\n" );
document.write( "* Since these choices are independent, we multiply the number of ways for each: 10 * 3 = 30\r<BR>\n" );
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document.write( "**d) 1 easy and 3 difficult routines:**\r<BR>\n" );
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document.write( "* There are 2 easy routines, and the student must choose 1: ²C₁ = 2! / (1! * 1!) = 2<BR>\n" );
document.write( "* There are 3 difficult routines, and the student must choose 3: ³C₃ = 3! / (3! * 0!) = 1<BR>\n" );
document.write( "* Since these choices are independent, we multiply the number of ways for each: 2 * 1 = 2<BR>\n" );
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