document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210229>Question 1210229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of subsets of<BR>\n" );
document.write( "S = \{1, 3, 8, 17, 30, 36, 47, 58\},<BR>\n" );
document.write( "so that the sum of the elements in the subset is less than 20. (Note that for the empty subset, we take the sum of the elements as 0.)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851694 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=851694\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solution:<BR>\n" );
document.write( "We are asked to find the number of subsets of $S = \{1, 3, 8, 17, 30, 36, 47, 58\}$ such that the sum of the elements in the subset is less than 20.\r<BR>\n" );
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document.write( "We can consider each element of $S$ and decide whether to include it in a subset. Since the elements are sorted, we can proceed by considering smaller elements first.\r<BR>\n" );
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document.write( "Case 1: Subsets with 0 elements<BR>\n" );
document.write( "The only subset with 0 elements is the empty set, $\emptyset$. The sum of its elements is 0, which is less than 20.<BR>\n" );
document.write( "Number of subsets = 1 ($\emptyset$)\r<BR>\n" );
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document.write( "Case 2: Subsets with 1 element<BR>\n" );
document.write( "The subsets with 1 element are $\{1\}, \{3\}, \{8\}, \{17\}, \{30\}, \{36\}, \{47\}, \{58\}$.<BR>\n" );
document.write( "The sums of the elements are 1, 3, 8, 17, 30, 36, 47, 58.<BR>\n" );
document.write( "The sums less than 20 are 1, 3, 8, 17.<BR>\n" );
document.write( "The subsets are $\{1\}, \{3\}, \{8\}, \{17\}$.<BR>\n" );
document.write( "Number of subsets = 4\r<BR>\n" );
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document.write( "Case 3: Subsets with 2 elements<BR>\n" );
document.write( "We consider pairs of distinct elements from $S$ and check if their sum is less than 20.<BR>\n" );
document.write( "Pairs involving 1: $\{1, 3\} \rightarrow 4$, $\{1, 8\} \rightarrow 9$, $\{1, 17\} \rightarrow 18$. (3 subsets)<BR>\n" );
document.write( "Pairs involving 3 (excluding pairs with 1): $\{3, 8\} \rightarrow 11$, $\{3, 17\} \rightarrow 20$ (sum not less than 20). (1 subset)<BR>\n" );
document.write( "Pairs involving 8 (excluding pairs with 1 and 3): $\{8, 17\} \rightarrow 25$ (sum not less than 20). (0 subsets)\r<BR>\n" );
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document.write( "The 2-element subsets with a sum less than 20 are $\{1, 3\}, \{1, 8\}, \{1, 17\}, \{3, 8\}$.<BR>\n" );
document.write( "Number of subsets = 4\r<BR>\n" );
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document.write( "Case 4: Subsets with 3 elements<BR>\n" );
document.write( "We consider triplets of distinct elements from $S$ and check if their sum is less than 20.<BR>\n" );
document.write( "Triplets involving 1 and 3: $\{1, 3, 8\} \rightarrow 12$, $\{1, 3, 17\} \rightarrow 21$ (sum not less than 20). (1 subset)<BR>\n" );
document.write( "Triplets involving 1 and 8 (excluding triplets with 3): $\{1, 8, 17\} \rightarrow 26$ (sum not less than 20). (0 subsets)\r<BR>\n" );
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document.write( "The 3-element subset with a sum less than 20 is $\{1, 3, 8\}$.<BR>\n" );
document.write( "Number of subsets = 1\r<BR>\n" );
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document.write( "Case 5: Subsets with 4 or more elements<BR>\n" );
document.write( "The smallest 4 elements of $S$ are 1, 3, 8, 17. Their sum is $1 + 3 + 8 + 17 = 29$, which is not less than 20. Therefore, there are no subsets with 4 or more elements whose sum is less than 20.\r<BR>\n" );
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document.write( "Total number of subsets with a sum less than 20 is the sum of the number of subsets from each case:<BR>\n" );
document.write( "Total = (Number of 0-element subsets) + (Number of 1-element subsets) + (Number of 2-element subsets) + (Number of 3-element subsets)<BR>\n" );
document.write( "Total = $1 + 4 + 4 + 1 = 10$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{10}$ </SPAN>\n" );
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