Question 1204712
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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?
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        In my solution, I will assume that the person "I" does not participate in any group of 14 persons.


        The problem is silent about it, so I can make such assumption.



<pre>
The number of all possible different groups of 14 members is  {{{C[n]^14}}}.


The problem asks you to find the minimal n such that

    {{{C[n-1]^14}}} <= 100, but {{{C[n]^14}}} > 100.


Obviously, n should be not less than 14.


The table below represnts several values of n >= 14 and relevant values of {{{C[n]^14}}}

     n          {{{C[n]^14}}}
 -----------------------------
    14		  1
    15		 15
    16		120


From the table, you may see that the minimal n is 16.


<U>ANSWER</U> to the problem's question is this: the smallest positive integer  number n satisfying the problem's requirement is 16.
</pre>

Solved.