We will of course use the graph theoretical translation. Let $a_k$ be the number of vertices of degree $36-k$, for $1\leq k \leq 36$. It is clear that under the given conditions $a_k \leq k$ for all $k$. It means the largest sum of degrees occurs when $a_k=k$ for $1\leq k \leq 8$ and $a_k = 0$ for $9\leq k \leq 36$, when $\sum_{k=1}^8 a_k = \sum_{k=1}^8 k = 36$, and the total number of edges is $\dfrac {1}{2} \sum_{k=1}^8 a_k(36 - a_k) = \dfrac {1}{2} \sum_{k=1}^8 k(36 - k) = 546$. A maximal model is the $8$-partite graph made by a partition of the $36$ vertices into $8$ classes, of cardinalities from $1$ to $8$, with no edges within any class, but all possible edges between different classes.

The answer is $3885$.But,I don't how it is possible.

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Olympus_mountaineer wrote: The answer is $3885$.But,I don't how it is possible.

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My answer was wrong.Actual answer is here

The 1st participant handshakes with rest 35 participants. Then 2nd participant handshakes with rest participants except 1st participant and last participant. That means he handshakes with 33 participants. Similarly next participant don't handshakes with 1st two and last participants that means 31 handshakes. So the series should be like this, 1+3+5+...+35=18²=324

Olympus_mountaineer wrote: Olympus_mountaineer wrote: The answer is $3885$.But,I don't how it is possible.

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My answer was wrong.Actual answer is here

how 84 accounts for no. of handshakes not occured .please tell in brief . i'm unable to understand the condition of question (no two participants with the same number of handshakes made, had shaken hands with each other)