document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208798>Question 1208798</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A and B be two sets containing 4 and 2 elements respectively. Then the number of subsets of the set AxB, each having at least three elements is<BR>\n" );
document.write( "A. 219<BR>\n" );
document.write( "B. 228<BR>\n" );
document.write( "C. 235<BR>\n" );
document.write( "D. 256 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847248 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=847248\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Construct two example sets<BR>\n" );
document.write( "A = {a, b, c, d}<BR>\n" );
document.write( "B = {e, f}\r<BR>\n" );
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document.write( "A x B = <a  rel=nofollow HREF=\"https://mathworld.wolfram.com/CartesianProduct.html\">cartesian product</a><BR>\n" );
document.write( "A x B = set of ordered pairs of form (p,q) where p is from set A and q is from set B<BR>\n" );
document.write( "A x B = {<BR>\n" );
document.write( "(a,e), (a,f),<BR>\n" );
document.write( "(b,e), (b,f),<BR>\n" );
document.write( "(c,e), (c,f),<BR>\n" );
document.write( "(d,e), (d,f)<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "If you want you can flatten the set into one single line like this<BR>\n" );
document.write( "A x B = { (a,e), (a,f), (b,e), (b,f), (c,e), (c,f), (d,e), (d,f) }<BR>\n" );
document.write( "But it might be preferable to have it broken up into rows.\r<BR>\n" );
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document.write( "4 items in set A, 2 items in set B, n = 4*2 = 8 ordered pairs in AxB.<BR>\n" );
document.write( "There are 2^n = 2^8 = 256 possible subsets of AxB (see <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Power_set\">Power Set</a>)<BR>\n" );
document.write( "This includes the set AxB itself and the empty set.<BR>\n" );
document.write( "Refer to Pascal's Triangle. Look at the row that starts with \"1,8,28...\". Each value in Pascal's Triangle can be computed using the nCr combination formula.<BR>\n" );
document.write( "All of those values in that row add to 256. \r<BR>\n" );
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document.write( "The first three items of this row correspond to:<ul><li>1 empty set</li><li>8 subsets that are <a  rel=nofollow HREF=\"https://mathworld.wolfram.com/SingletonSet.html\">singletons</a>. These are the subsets that have exactly one ordered pair.</li><li>28 subsets that have two ordered pairs</li></ul>Add up those first three values to get 1+8+28 = 37<BR>\n" );
document.write( "There are 37 subsets with at most 2 ordered pairs.<BR>\n" );
document.write( "Then 256-37 = <font color=red>219</font> is the amount of subsets that have between 3 ordered pairs and 8 ordered pairs.\r<BR>\n" );
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document.write( "Answer: <font color=red>Choice A</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );