# Is it $fact$$or$$fiction$ ? Is there any relation between both the given facts or this is coincidence : sum of $n$ natural numbers i.e. $\frac { n(n+1) }{ 2 }$ and ${n+1 \choose 2}$ Note by Rishabh Jain
6 years, 5 months ago

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Yes There is relation between both If n(n+1)/2n When we have to find mean Then its equal to n+1/2

- 6 years, 5 months ago

- 6 years, 5 months ago

- 6 years, 5 months ago

Okay.

- 6 years, 5 months ago

- 6 years, 5 months ago

Yeah, I know. But he's looking for a reason aside from the coincidence, like a geometric or combinatorial proof. :P

- 6 years, 5 months ago

exactly !!

- 6 years, 5 months ago

No, I quite literally mean you answered the command given by Rishabh.

- 6 years, 5 months ago

If you factor $n+1$ choose $2$ you get $\dfrac{n(n+1)}{2!}$.

- 6 years, 5 months ago

oh! according to my knowledge $2!$$=$$2$ so what is the relation between the two

- 6 years, 5 months ago

Imagine the first one in a line of (n+1) persons saying "Hello" to all the n, second saying to remaining (n-1), third to rest of (n-2) and so on till the last but one says "Hello' to the last one. Total number of "Hello' is a combination term, 2 out of (n+1) as each and every pair said "Hello", which is also the sum of first n natural numbers.

- 6 years, 5 months ago