document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210202>Question 1210202</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851592 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=851592\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Understand the Constraints**\r<BR>\n" );
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document.write( "* We are looking for subsets of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.<BR>\n" );
document.write( "* Each subset must have at least two elements.<BR>\n" );
document.write( "* The sum of the two largest elements in each subset must be 4.\r<BR>\n" );
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document.write( "**2. Identify the Possible Pairs**\r<BR>\n" );
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document.write( "Since the sum of the two largest elements must be 4, the possible pairs of largest elements are:\r<BR>\n" );
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document.write( "* (0, 4)<BR>\n" );
document.write( "* (1, 3)<BR>\n" );
document.write( "* (2, 2)\r<BR>\n" );
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document.write( "**3. Analyze Each Pair**\r<BR>\n" );
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document.write( "* **Pair (0, 4):**<BR>\n" );
document.write( "    * The subset must contain 0 and 4.<BR>\n" );
document.write( "    * Other elements can be any subset of {}.<BR>\n" );
document.write( "    * There is only one subset: {0, 4}.\r<BR>\n" );
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document.write( "* **Pair (1, 3):**<BR>\n" );
document.write( "    * The subset must contain 1 and 3.<BR>\n" );
document.write( "    * Other elements can be any subset of {0}.<BR>\n" );
document.write( "    * Possible subsets: {1, 3}, {0, 1, 3}.<BR>\n" );
document.write( "    * There are 2 subsets.\r<BR>\n" );
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document.write( "* **Pair (2, 2):**<BR>\n" );
document.write( "    * The subset must contain 2.<BR>\n" );
document.write( "    * Other elements can be any subset of {0, 1}.<BR>\n" );
document.write( "    * Possible subsets: {2, 2} which is just {2}, {0, 2}, {1, 2}, {0, 1, 2}.<BR>\n" );
document.write( "    * However, we need at least two elements, so we must exclude {2}.<BR>\n" );
document.write( "    * Possible subsets: {0, 2}, {1, 2}, {0, 1, 2}.<BR>\n" );
document.write( "    * There are 3 subsets.\r<BR>\n" );
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document.write( "**4. Count the Subsets**\r<BR>\n" );
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document.write( "* For (0, 4): 1 subset<BR>\n" );
document.write( "* For (1, 3): 2 subsets<BR>\n" );
document.write( "* For (2, 2): 3 subsets\r<BR>\n" );
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document.write( "Total number of subsets = 1 + 2 + 3 = 6\r<BR>\n" );
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document.write( "**Therefore, there are 6 subsets of {0, 1, ..., 9} that have the property that there are at least two elements and the sum of the two largest elements is 4.**<BR>\n" );
document.write( " </SPAN>\n" );
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