Sum of combinations of sets

Question

Consider all $1000$–element subsets of $\{1,2,3,4,\dots 2015\}$. From each such subset select least element. Find Arithmetic Mean of all these elements.

I easily managed the trick but my approach got me a lengthy answer. I selected leaste numbers and multiplied them by their number of subsets possible such as; for $1$, number of subsets possible$=1016$, for $2$, $1015$ and so on. By adding this in series($1\times 1016 + 2\times 1015 + \dots 1016\times 1$) we would get the answer but it got to be a lengthy one.

Answer 1

This is not a complete solution, but I HTH.

If the least element is $1$ then the rest of a subset is $999$ elements from remaining $2014$ elements, so there are $\binom {2014}{999}$ such sets.
For the least element $2$ the rest of a subset is $999$ elements out of $2013$, hence $\binom {2013}{999}$.
And so on.
The maximum minimal element is $1016$, in which case there is just one subset possible: $\{1016,\dots 2015\}$.

Eventually your answer is $\sum_{i=1}^{1016}\binom{2015-i}{999}$,
or equivalently $\sum_{s=999}^{2014}\binom s{999}.$

Answer 2

Following the same line as in answer of CiaPan the final result is:

$\langle N\rangle={\binom{2015}{1000}}^{-1} \sum_{i=1}^{1016}i \binom{2015-i}{999}=\frac{288}{143}\approx 2,$

where ${\binom{A}{B}}$ means binomial coefficient.

Answer 3

Well the answer to this question is simply $\frac {2016}{1001}$

For the generalised method for such question refer to my answer