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### Black.n.White's blog

By Black.n.White, history, 7 months ago, , My friend gave me a problem, which can be reduced to the following.

Let $S(k)$ be the set of all arrays of size $n$ that contains $k$ ones and $n - k$ zeroes. Let for some array $s \in S(k)$ , $pos_s[1..k]$ denotes the position of those ones (say in increasing order, it doesn't matter actually). We have to calculate.

$\sum_{s \in S(k)} {\sum_{i = 1}^{k} pos_s[i] }$

i.e, for all arrays $s \in S(k)$ we have to sum the positions of all $k$ ones.

My thoughts : Modification to binomial expansion? Noo. hmm, what about counting how many times each position occurs, but then there will be intersections, like I might put two ones at the same index, so ? what about inclusion-exclusion? No it's getting a lot messy.

Linearity of Expectation : Let's find the expected value of $\sum_{i = 1}^{k} pos_s[i]$

$E[\sum_{i = 1}^{k} pos_s[i]]$

Let's look at it it some other way. We choose random variables $X_{1, 2, .. k}$ lying in $[1, n]$ with same probability (Note that I'm not saying they're independent. ). Thus $E[X_i] = \frac{1 + 2 + .. n}{n} = \frac{n + 1}{2}$ Now by linearity of expectation,

$E[\sum_{i = 1}^{k} X_i] = \sum_{i = 1}^{k} E[X_i] = \sum_{i = 1}^{k} \frac{n + 1}{2} = \frac{k(n + 1)}{2}$

.

Now, it's left to you to realise that

$E[\sum_{i = 1}^{k} pos_s[i] = E[\sum_{i = 1}^{k} X_i]$

.

And since we've $n \choose k$ arrays possible, we deduce,

$\sum_{s \in S(k)} {\sum_{i = 1}^{k} pos_s[i] }= \frac{{n \choose k} k (n + 1)}{2}$

.

I went numb for a few seconds after completing the solution, I mean wtf, what about intersection, how can this be even possible, you can't just choose it randomly (I know the formal proof of LOE, but still :\ )

There are few things, that I know are true, but I can never digest them, "My favourite song has just 7 million views on youtube" is on second", "Linearity of expectation actually works" remains at first.

Maybe I overkilled it. Other easier solutions are welcome :) By Black.n.White, history, 9 months ago, , I'm getting WA on codeforces, while its working correctly for that case on my system and even on codeforces custom test. Also the checkers comment is too weird.

Can't find file K:\invoker-prod\work\codeforces6\25bd73a20640963248bd7f3788f1424e\check-1b3eb285945d2168cb2ee5ad536c2f21\run\output.txt [invokerId=7b1ec022e70f30d35dabadbce49e1a1e, location=203279216] (1).

Anybody knows what this might mean? code

By Black.n.White, history, 10 months ago, , I am getting a very strange bug in my code for today's round problems B1, 2.

At a moment, I'm having a set of pairs.

[s]: {(1, 1), (1, 4), (1, 5), (1, 9), (1, 10), (2, 2), (2, 7)}

Then I'm accessing the last and second last element. (line 51-54 in mycode)

it = s.rbegin();  // prints *it = (2, 7)
it--;
// prints *it = (1, 10)


But for some weird reason it jumps off (2, 2) and I'm unable to figure out why.

Any help will be appreciated. #c++,
By Black.n.White, history, 18 months ago, ,  