MHB What is the Minimum Number of Friends Needed for Unique Dinner Invitations?

[Lancelot1]

Hello all,

I am trying to solve this one:

John has n friends . He wants to invite in each evening (365 days a year) three of his friends for dinner. What should be the size of n, such that it will be possible not to invite the same triplet twice ?

What I did was:

\[\binom{n}{3}\leq 365\]

which turns into:

\[\frac{(n-1)(n-2)n}{6}\leq 365\]

I have tried to solve it, manually and with a mathematical software, in both ways n was not a natural number...where is my mistake ?

 

[Evgeny.Makarov]

I have tried to solve it, manually and with a mathematical software, in both ways n was not a natural number

First, the inequality should be $\binom{n}{3}\ge365$. Second, the answer to this problem is an inequality $n\ge\ldots$, not a specific value of $n$. For the lower bound of $n$ take the smallest natural number that is equal to or larger than the root of that equation. Note that $\binom{14}{3}=364$.