Variance for summing over distinct random integers Let $L=\{a_1,a_2,\ldots,a_k\}$ be a random (uniformly chosen) subset of length $k$ of the numbers $\{1,2,\ldots,n\}$. I want to find $\operatorname{Var}(X)$ where $X$ is the random variable that sums all numbers with $k < n$.
Earlier today I asked about the expected value, which I noticed was easier than I thought. But now I am sitting on the variance since several hours but cannot make any progress. I see that $E(X_i)=\frac{n+1}{2}$ and $E(X)=k \cdot \frac{n+1}{2}$, I tried to use $\operatorname{Var}\left(\sum_{i=1}^na_iX_i\right)=\sum_{i=1}^na_i^2\operatorname{Var}(X_i)+2\sum_{i=1}^{n-1}\sum_{j=i+1}^na_ia_j\operatorname{Cov}(X_i,X_j)$ but especially the second sum is hard to evaluate by hand ( every time I do this I get a different result :-) ) and I have no idea how to simplify the Covariance term. Furthermore I know that $\operatorname{Var}(X)=\operatorname{E}\left(\left(X-\operatorname{E}(X)\right)^2\right)=\operatorname{E}\left(X^2\right)-\left(\operatorname{E}(X)\right)^2$, so the main Problem is getting $=\operatorname{E}\left(X^2\right)$. Maybe there is also a easier way than to use those formulas.
I think I got the correct result via trial and error: $\operatorname{Var}(X)=(1/12) k (n - k) (n + 1)$ but not the way how to get there..

Let me (almost) quote myself:
Hint: (1) Compute $E(a_1^2)$. (2) Show that $E(a_k^2)=E(a_1^2)$ for every $k$. (3) Compute $E(a_1a_2)$. (4) Show that $E(a_ka_i)=E(a_1a_2)$ for every $k\ne i$. (5) Deduce $E(X^2)$.
You probably already know the distribution of $a_1$ hence (1) and (2) are easy. Now, you simply need to know the distribution of $(a_1,a_2)$.
This is (still) a good example where computing the distribution is messy but computing expectations (and variances) is easy.
Edit One finds 
$$
\mbox{var}(a_1)=\frac{1}{12}(n+1)(n-1),\quad \mbox{cov}(a_1,a_2)=-\frac{1}{12}(n+1).
$$
Hence
$$
\mbox{var}(X)=k\mbox{var}(a_1)+k(k-1)\mbox{cov}(a_1,a_2)=\frac{1}{12}k(n+1)(n-k),
$$
as computed by the OP. Sanity check: if $k=n$, $\mbox{var}(X)=0$.
