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I have n friends. Every night of the 365-day year I invite three of them to dinner. What is the smallest n could be such that it is still possible for me to make these invitations without ever inviting the same group of three friends? (I can invite the same friend, or even the same pair of friends, but never the same whole group of three.)

 Jul 9, 2021
 #1
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We  need to  find  this

 

C ( n , 3)    =  365

 

        n!

__________  =     365

(n - 3)!  3!

 

 

(n) ( n - 1) (n - 2)  =  365 * 3!

 

n (n^2  - 3n  + 2)  =   2190

 

n^3 - 3n^2   + 2n  -  2190  =   0

 

Using a little technology, n   ≈  14.012

 

So....you would  need    15  friends

 

P.S. ....if  there were only  364  nights  you could invite exactly 14  friends   !!!

 

cool cool cool

 Jul 9, 2021
 #2
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Slick...Btw thanks!

Guest Jul 9, 2021

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