document.write( "Question 755607: 36. If the number of subsets with 4 elements of a set A is equal to the number of
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Algebra.Com's Answer #459781 by Edwin McCravy(20055)\"\" \"About 
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36. If the number of subsets with 4 elements of a set A is equal to the number of
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document.write( "One solution would come from the case that if A has 3 or fewer elements, there\r\n" );
document.write( "would be 0 subsets with 4 elements and 0 subsets with 5 elements, and 0 = 0. \r\n" );
document.write( "Let's see if there are any other solutions: \r\n" );
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document.write( "If the set contains N elements, then the number of subsets it has with K\r\n" );
document.write( "elements is C(N,K) = \"N%21%2F%28K%21%28N-K%29%21%29%29\"\r\n" );
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document.write( "Suppose set A has N elements, where integer N > 3.  Then we have\r\n" );
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document.write( "C(N,4) = C(N,5) \r\n" );
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document.write( "\"N%21%2F%284%21%28N-4%29%21%29\" = \"N%21%2F++%285%21%28N-5%29%21++%29\" \r\n" );
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document.write( "Cross multiply\r\n" );
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document.write( "N!5!(N-5)! = N!4!(N-4)!\r\n" );
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document.write( "Divide both sides by N!\r\n" );
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document.write( "5!(N-5)! = 4!(N-4)!\r\n" );
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document.write( "Write out or indicate all the factors of all the factorial:\r\n" );
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document.write( "5·4·3·2·1(N-5)(N-4)(N-3)···1 =  4·3·2·1(N-4)(N-3)···1 \r\n" );
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document.write( "Divide both sides through by all the common factors, and we have:\r\n" );
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document.write( "5(N-5) = 1\r\n" );
document.write( " 5N-25 = 1\r\n" );
document.write( "    5N = 26\r\n" );
document.write( "     N = \"26%2F5\" = \"5%261%2F5\"\r\n" );
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document.write( "That is not an integer, so there are no other cases.\r\n" );
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document.write( "So A has fewer than 4 elements.  If A has 3 elements, there is 1\r\n" );
document.write( "subset, the improper subset A). If A has 2,1,or 0 elements, then \r\n" );
document.write( "the answer is 0.\r\n" );
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document.write( "Answer: The number of subsets with 3 elements is 0 or 1, or \"None of the above\".\r\n" );
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document.write( "Edwin
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