document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177694>Question 1177694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.) A dance instructor asks each student to do 4 out of 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  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850405 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=850405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve each part of the problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**Understanding Combinations**\r<BR>\n" );
document.write( "\n" );
document.write( "We'll use combinations (nCr) to calculate the number of ways to choose routines. The formula for combinations is:\r<BR>\n" );
document.write( "\n" );
document.write( "* nCr = n! / (r! * (n-r)!)\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* n = total number of items<BR>\n" );
document.write( "* r = number of items to choose\r<BR>\n" );
document.write( "\n" );
document.write( "**A. 4 Moderately Difficult Routines**\r<BR>\n" );
document.write( "\n" );
document.write( "* Total moderately difficult routines: 5<BR>\n" );
document.write( "* Routines to choose: 4\r<BR>\n" );
document.write( "\n" );
document.write( "* Number of ways = 5C4 = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = 5\r<BR>\n" );
document.write( "\n" );
document.write( "**B. 4 Easy or Moderately Difficult Routines**\r<BR>\n" );
document.write( "\n" );
document.write( "* Total easy routines: 2<BR>\n" );
document.write( "* Total moderately difficult routines: 5<BR>\n" );
document.write( "* Total easy or moderately difficult routines: 2 + 5 = 7<BR>\n" );
document.write( "* Routines to choose: 4\r<BR>\n" );
document.write( "\n" );
document.write( "* Number of ways = 7C4 = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35\r<BR>\n" );
document.write( "\n" );
document.write( "**C. 2 Moderately Difficult and 2 Difficult Routines**\r<BR>\n" );
document.write( "\n" );
document.write( "* Moderately difficult routines: 5<BR>\n" );
document.write( "* Difficult routines: 3\r<BR>\n" );
document.write( "\n" );
document.write( "1.  Choose 2 moderately difficult routines: 5C2 = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10<BR>\n" );
document.write( "2.  Choose 2 difficult routines: 3C2 = 3! / (2! * 1!) = 3\r<BR>\n" );
document.write( "\n" );
document.write( "* Total number of ways = 10 * 3 = 30\r<BR>\n" );
document.write( "\n" );
document.write( "**D. 1 Easy and 3 Difficult Routines**\r<BR>\n" );
document.write( "\n" );
document.write( "* Easy routines: 2<BR>\n" );
document.write( "* Difficult routines: 3\r<BR>\n" );
document.write( "\n" );
document.write( "1.  Choose 1 easy routine: 2C1 = 2! / (1! * 1!) = 2<BR>\n" );
document.write( "2.  Choose 3 difficult routines: 3C3 = 3! / (3! * 0!) = 1\r<BR>\n" );
document.write( "\n" );
document.write( "* Total number of ways = 2 * 1 = 2\r<BR>\n" );
document.write( "\n" );
document.write( "**Answers**\r<BR>\n" );
document.write( "\n" );
document.write( "* **A. 4 moderately difficult routines:** 5 ways<BR>\n" );
document.write( "* **B. 4 easy or moderately difficult routines:** 35 ways<BR>\n" );
document.write( "* **C. 2 moderately difficult and 2 difficult routines:** 30 ways<BR>\n" );
document.write( "* **D. 1 easy and 3 difficult routines:** 2 ways<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );