document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250032>Question 250032</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 37. In this exercise we study the connection between sets (from Chapter 7) and combinations<BR>\n" );
document.write( "(from Chapter 8).<BR>\n" );
document.write( "a. Given a set with n elements, what is the number of subsets of size 0? of size 1? of<BR>\n" );
document.write( "size 2? of size n?<BR>\n" );
document.write( "b. Using your answer from part a, give an expression for the total number of subsets of<BR>\n" );
document.write( "a set with n elements.<BR>\n" );
document.write( "c. Using your answer from part b and a result from Chapter 7, explain why the following<BR>\n" );
document.write( "equation must be true: (n) + (n) + (n) + ... + (n) =2n<BR>\n" );
document.write( "                       (0)   (1)   (2)         (n)<BR>\n" );
document.write( "d. Verify the equation in part c for n=4 and n=5<BR>\n" );
document.write( "e. Explain what the equation in part c tells you about Pascal’s triangle </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #185510 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=185510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. Given a set with n elements, what is the number of subsets <BR>\n" );
document.write( "of size 0?---------nC0 = 1 <BR>\n" );
document.write( "of size 1? of------nC1 = n<BR>\n" );
document.write( "size 2?------------nC2= (n(n-1))/2 <BR>\n" );
document.write( "of size n?---------nCn = 1<BR>\n" );
document.write( "----------------------------------------------<BR>\n" );
document.write( "b. Using your answer from part a, give an expression for the total number of subsets of a set with n elements.--------------<BR>\n" );
document.write( "Total number of subsets = nC0 + nC1 + nC2 + ... + nCn = 2^n<BR>\n" );
document.write( "----------------------------------------<BR>\n" );
document.write( "c. Using your answer from part b and a result from Chapter 7, explain why the following<BR>\n" );
document.write( "equation must be true: nC0 + nC1 + nC2 + ... + nCn = 2^n<BR>\n" );
document.write( "Ans: Every element in the set of \"n\" is either chosen or not<BR>\n" );
document.write( "chosen in each subset.  So the number of possible subsets<BR>\n" );
document.write( "is 2^n  (a choice with 2 possible outcomes made n times)<BR>\n" );
document.write( "------------------------------------------------------------------<BR>\n" );
document.write( "d. Verify the equation in part c for n=4 and n=5<BR>\n" );
document.write( "4C0 + 4C1 + 4C2 + 4C3 + 4C4 <BR>\n" );
document.write( "= 1 + 4 + 6 + 4 + 1<BR>\n" );
document.write( "= 16<BR>\n" );
document.write( "= 2^4<BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( "Note: I'll leave n = 5 to you<BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( "e. Explain what the equation in part c tells you about Pascal’s triangle<BR>\n" );
document.write( "The sum of the elements in the kth row is 2^(k-1)<BR>\n" );
document.write( "====================================================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );