document.write( "Question 1204712: I run a book club with $n$ people, not including myself. Every day, for $100$ days, I invite $14$ members in the club to review a book. What is the smallest positive integer $n$ so that I can avoid ever having the exact same group of $14$ members over all $100$ days? \n" ); document.write( "

Algebra.Com's Answer #841152 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "I run a book club with n people, not including myself. Every day, for 100 days, I invite 14 members in the club
\n" ); document.write( "to review a book. What is the smallest positive integer n so that I can avoid ever having the exact same group
\n" ); document.write( "of 14 members over all 100 days?
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\n" ); document.write( "\n" ); document.write( " In my solution, I will assume that the person \"I\" does not participate in any group of 14 persons.\r
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\n" ); document.write( "\n" ); document.write( " The problem is silent about it, so I can make such assumption.\r
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document.write( "The number of all possible different groups of 14 members is  .\r\n" );
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document.write( "The problem asks you to find the minimal n such that\r\n" );
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document.write( "     <= 100, but  > 100.\r\n" );
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document.write( "Obviously, n should be not less than 14.\r\n" );
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document.write( "The table below represnts several values of n >= 14 and relevant values of \r\n" );
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document.write( "     n          \r\n" );
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document.write( "    14		  1\r\n" );
document.write( "    15		 15\r\n" );
document.write( "    16		120\r\n" );
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document.write( "From the table, you may see that the minimal n is 16.\r\n" );
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document.write( "ANSWER to the problem's question is this: the smallest positive integer  number n satisfying the problem's requirement is 16.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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