Wolfgang thanks for the question ... LOL ! This one was hard to crack ! I had to make a few assumptions to get it right. I would like to have your feedback on this one, because the derivation is tricky.
I hope the explanation will be crystal clear.
This calculation will help to realize the concept of communication channel.
Say, there are “n” Persons and they are having communication with each other. We need to find how many communication channels are possible here.
Take the case of the first person. He can communicate with rest of the (n-1) person. So, here are (n-1) communication channels.
The Second person can communicate with ((n-1)-1) person because his communication with the first person is already taken into account for the calculation of first person. So, here are (n-2) communication channels.
Similarly, the third person can communicate with ((n-1)-2) because his communication with the first and second person is already taken into account for the calculation of first and second person. So, it’s (n-3) channels.
Like that, if we proceed, the total number of Channels used by n person, will be = (n-1)+(n-2)+(n-3)+......+1 = 1+2+3+.....+(n-1)
We have a sum of a series for "(n-1)" number whose first number is "1" and the common difference is also "1".
The Formula for Sum of a series is: S = n[2a+(n-1)d]/2
(where, n = total numbers, a = first number, d = common difference)
So, it gives us the Total Communication Channel
=(n-1)*[2*1+((n-1)-1)*1]/2 = (n-1)*[2+(n-2)]/2 = n*(n-1)/2
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