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# EASYEX - Editorial

Author: Pavel Sheftelevich
Tester: Mugurel Ionut Andreica
Editorialist: Lalit Kundu

Hard

# PROBLEM:

Suppose you have a dice with $K$ faces with numbers from $1$ to $K$ written on it, and integers $L$ and $F(0 \lt L \le K)$. You roll it $N$ times. Let $a_i$ be the number of times(out of the $N$ rolls) that a face with number $i$ written on it came up as the top face of the dice. Find the mathematical expectation of the value $a_1^F * a_2^F * ... * a_L^F$.

# QUICK EXPLANATION:

================
Write $a_i$ as $x_{i, 1} + x_{i, 2} + ... x_{i, N}$, where $x_{i, j}$ is $0$ or $1$ depending on whether we got the value $i$ on the $j^{th}$ dice roll. We can write probability of $x_{i, j}$ being $1$ as $\frac{1}{K}$ as we can get any value from $1$ to $K$ equally probably in the $j^{th}$ roll of dice.

For $F=1$, our answer will be number of terms of form $x_{1, A_1} * x_{2, A_2} * ... * x_{L, A_L}$ where all $A_i$ are distinct, divided by $K^L$. Number of such terms is $\binom{N}{L} * L!$.

For $F>1$, for each polynomial of form $(x_{1, 1} + x_{1, 2} + ... x_{1, N})^F$ first we individually define $\textrm{num_ways}[F, i]$ as number of ways to choose $F$ variables $f_1, f_2 ... f_F$ such that $1 \le f_j \le i$ and and atleast one of the $f_j$ is $i$. Then, we traverse over number of terms $P$ from $L$ to $L*F$ and for each $P$ we add to answer $\frac{\textrm{num}}{\textrm{den}}$ where $\textrm{num} = \binom{N}{P} * P! * n_P$ and $\textrm{den} = K^P$; where $n_p$ can be written as
coefficient of $x^P$ in polynomial $(\textrm{num_ways}[F, 0]x^0 + \textrm{num_ways}[F, 1]x^1 + \textrm{num_ways}[F, 2]x^2 + ... + \textrm{num_ways}[F, F]x^F)^L$.

# EXPLANATION:

================

Warning
I personally found it a very complex mathematical problem, but I am not a pure maths guy, so it is fine. Now, if you really want to understand this and you are not a math guy, you'll need to put down your utmost attention to the detail in this editorial.

Now, let us move to solving the problem.

First a very basic definition of the expected value $E[X]$. Suppose random variable $X$ can take value $x_1$ with probability $p_1$, value $x_2$ with probability $p_2$, and so on, up to value $x_k$ with probability $p_k$. Then the expectation of this random variable $X$, $E[X]$ is defined as $x_1p_1 + x_2p_2 + ... + x_kp_k$.

The first most basic observation is that we can't directly deal with variables $a_i$, we need to break them down into smaller components. The most easy way to deal with them would be to write $a_i$ as $x_{i, 1} + x_{i, 2} + ... x_{i, N}$, where $x_{i, j}$ is $0$ or $1$ depending on whether we got the value $i$ on the $j^{th}$ dice roll. We can write probability of $x_{i, j}$ being $1$ as $\frac{1}{K}$ as we can get any value from $1$ to $K$ equally probably in the $j^{th}$ roll of dice.

## F=1

First, let's solve for the case where $F=1$. We are interested in
$E[(x_{1, 1} + x_{1, 2} + ... x_{1, N})*(x_{2, 1} + x_{2, 2} + ... x_{2, N})* ... *(x_{L, 1} + x_{L, 2} + ... x_{L, N})]$.
Now, this polynomial inside can be expanded to get various terms of the form $x_{1, A_1} * x_{2, A_2} * ... * x_{L, A_L}$ where $1 \le A_i \le N$. Our expected value is sum of expected value of all these terms. Now, all these terms $x_{i, A_i}$ should be $1$ for this certain value to contribute to the expected value. Now, thing worth noting here is that if in a particular term for any $p$ and $q$ such that $1 \le p \ne q \le L$ and $A_p = A_q$, the expected value of this term is zero because probability of such an event is zero. For example, for a term like $x_{1, 2} * x_{2, 2} * ...$, the probability of this term is zero because this the term is stating that we got value $1$ in $2^{nd}$ dice roll and we also got value $2$ in $2^{nd}$ dice roll which is not possible.

So, we are only interested in terms where all $A_i$ are distinct. Probability of each such term to be $1$ is $\frac{1}{K^L}$, because we need each term to be $1$. So, if there are $n$ variables of form $x_{1, A_1} * x_{2, A_2} * ... * x_{L, A_L}$ where all $A_i$ are distinct, our answer is $\frac{n}{K^L}$(this follows from the definition of expectation). Let's see what is value of $n$. We are trying to get $L$ distinct values out of $N$ values, which is basically $\binom{N}{L}$ and note that order also matters here, so $n = \binom{N}{L} * L!$, where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is number of ways to select $k$ distinct objects out of $n$ distinct objects and $n! = 1*2*...*n$.

So, we say here that our numerator is $\frac{N!}{(N-L)!}$, where $N \le 10^9$ and $L \le 50,000$. We can write this as $(N-L+1) * (N-L+2) * ... * (N)$, whih can be calculated in $O(L)$. We can easily calculate denominator i.e $K^L$ here using modular exponention. For inverse modulo, since modulo is prime we can use Fermat's Theorem.

## F > 1

Before dwelling deep into the solution, I'd tell here an interesting result which we'll be using ahead. Understanding this is important before going into the solution. Let's say there are variables $x_1, x_2 ... x_N$ which are non-negative integers. For assigning an integer $x$ to a variable $x_i$, there are $f(x)$ number of ways to do it. Now, we want to calculate total number of ways to choose integers $x_1, x_2 ... x_N$ such that $x_i \ge 0$ are integers and $x_1 + x_2 + ... + x_N = M$. So, here we need to consider each integral solution of type $y_1, y_2, ... y_N$, such that $y_1 + y_2 + ... + y_N = N$, and for each such integral solution add to answer $f(y_1) * f(y_2) * ... * f(y_N)$.

I won't derive the result here; total number of ways in this case are coefficient of $x^M$ in polynomial $(f(0)x^0 + f(1)x^1 + f(2)x^2 + ... f(M)x^{M})^N$. You can read more about use of generating functions in combinatorics here.

Now, this is where it gets interesting. Here we are interested in
$E[(x_{1, 1} + x_{1, 2} + ... x_{1, N})^F * (x_{2, 1} + x_{2, 2} + ... x_{2, N})^F * ... *(x_{L, 1} + x_{L, 2} + ... x_{L, N})^F]$.
Now, what is different from $F=1$? In the earlier case, there were $L$ variables in each term, but now we can have $P$ variables where $P$ ranges from $L$ to $L*F$. For example, this is a term with $L*F$ variables
$x_{1, A_1} * x_{1, A_2} * ... * x_{1, A_F} *$
$x_{2, B_1} * x_{2, B_2} * ... * x_{2, B_F} *$
$.$
$.$
$.$
$x_{L, Z_1} * x_{L, Z_2} * ... * x_{L, Z_F}$.

where $1 \le A_i, B_i, ... Z_i \le N$.

And the observation is that if for any $i, j$ such that $1 \le i \le j \le F$, $A_i = B_j$ or $A_i = C_j$ or ... or $A_i = Z_j$ and similarly for $B$ and $C$ and so on, that particular term won't contribute anything to answer by the same logic as we had in case of $F=1$. However, any $A_i$ could be equal to $A_j$ as it just denotes that $1$ came in $A_i$ roll as well as $A_j$ roll of the dice.

So, what is our approach going to be? For each $P$ from $L$ to $L*F$, we'll calculate number of terms with $P$ variables. Say, if number of such terms is $n_P$, then we add to our answer $\frac{\textrm{num}}{\textrm{den}}$ where $\textrm{num} = \binom{N}{P} * P! * n_P$ and $\textrm{den} = K^P$ by the same logic as we had in $F=1$.

Now, let's see how to calculate $n_P$. First we should remember what $n_P$ is. It is number of terms in expansion of our overall polynomial such that there are $P$ variables in those terms.

There are $L$ individual polynomials $(x_{1, 1} + x_{1, 2} + ... x_{1, N})^F$, $(x_{2, 1} + x_{2, 2} + ... x_{2, N})^F$ and so on till $(x_{L, 1} + x_{L, 2} + ... x_{L, N})^F$ that we are interested in. We'll try to process each polynomial individually.

Let's say $W_i$ denotes the number of variables the $i^{th}$ polynomial of the total $L$ polynomials contribute. We want to calculate number of ways such that $W_1 + W_2 + ... + W_L = P$ where $W_i \ge 0$. But, for each value $\ge 0$ we give to any variable $W_i$, there are some number of ways attached with it too. For example, if we give $W_1$ a value of $1$, then we also have to multiply number of ways to choose $1$ distinct value $x(1 \le x \le N)$ such that $A_1 = A_2 = ... A_F = x$, or if we give a value of $2$ to $W_2$, then we have to choose $B_1, B_2 ... B_F$ from $1$ to $N$ in such a way that only $2$ distinct values occur in these.

Now, another interesting thing is that number of ways to choose $i$ distinct values of $1$ to $N$ doesn't depend on whether we are choosing those values for $W_1$ or $W_2$ and so on. So, if we define a new term here $\textrm{num_ways}[F, i]$ as number of ways to choose $F$ variables $f_1, f_2 ... f_F$ such that $1 \le f_j \le i$ and atleast one of the $f_j$ is $i$, then $n_p$ can be written as
coefficient of $x^P$ in polynomial $(\textrm{num_ways}[F, 0]x^0 + \textrm{num_ways}[F, 1]x^1 + \textrm{num_ways}[F, 2]x^2 + ... + \textrm{num_ways}[F, F]x^F)^L$.
This follows from the result written at starting of this section. Let's assume that we have calculated $\textrm{num_ways}[F, i]$ for all $0 \le i \le F$, then how do we proceed?

Note that our $P$ ranges from $L$ to $L*F$, so enumerating the whole polynomial
$(\textrm{num_ways}[F, 0]x^0 + \textrm{num_ways}[F, 1]x^1 + \textrm{num_ways}[F, 2]x^2 + ... + \textrm{num_ways}[F, F]x^F)^L$
is a good option. Now, raising polynomials to power can be done is same way as we raise integers to power via fast exponentiation. However, to quickly multiply two polynomials we'll use FFT. Using FFT we can multiply two polynomials of degree $N$ in $N \textrm{log}N$. So, our total complexity of evaluating that polynomial would be complexity of multiplying two polynomials of degree atmost $L*F$ atmost $O(\textrm{log} L)$ times(due to fast exponentiation). So, total complexity would be $O(L * F * \textrm{log}(L * F) * \textrm{log} L)$ which can written as $O(L * F * \textrm{log}(L * F)$.

But, but, there is an other interesting observation to avoid all this computation. Note that we multiply $n_p$ with $\binom{N}{P} * P!$ in numerator and divide by $K^P$ while adding to answer. But, our answer is modulo $2003$, so, for $P > 2003$, $\binom{N}{P} * P!$ is guaranteed to be zero (because it is the product of at least 2003 coefficients, one of which is certainly dvisible by 2003). So, we never need more than the first 2003 coefficients of the final polynomials.

Now, the part that we have left to understand is how to calculate $\textrm{num_ways}[F, i]$ for $1 \le i \le F$. To revise, $\textrm{num_ways}[F, i]$ is number of ways to choose $F$ variables $f_1, f_2 ... f_F$ such that $1 \le f_j \le i$ and and atleast one of the $f_j$ is $i$. We can easily define a recurrence $\textrm{num_ways}[F, i] = \textrm{num_ways}[F-1, i-1] + i*\textrm{num_ways}_[F-1, i]$.

How do we arrive at this reccurence?
The first term indicates the case that the value $i$ value occurs only once on $f_1, f_2, ..., f_F$, so we add number of ways to choose $i-1$ values for $F-1$ values and second term denotes the case that value $i$ has occured earlier, therefore it can have any of the $1, 2, ... i$ already existing values. So, we multiply $\textrm{num_ways}_[F-1, i]$ with $i$ and add to answer. Incidentally, these numbers are Stirling numbers of second kind. So, we can calculate all required values in $O(F^2)$ here.

## PROBLEMS TO SOLVE

===================

# AUTHOR'S, TESTER'S SOLUTIONS:

This question is marked "community wiki".

3.0k93164187
accept rate: 12%

19.8k350498541

 2 I will give a try to explain how the we end up with above formula, I will consider small case, and the result then can be generalized after few observations. I hope it will help you. answered 14 Jul '15, 14:03 5★avmnusng 147●3 accept rate: 9% 1 You might be wondering why it is considered (p1+p2+...+pL)^N, because it must have been considered something (p1+p2+...+pK)^N, the reason is when you perform the differentiation, the frequency count ai, for i>L, will make no change. You can easily verify this by replacing the (p1+p2)^N term in the explanation with (p1+p2+...+pK)^N, because this (p1+p2+...+pK)^N will be 1, as (p1+p2+...+pK) = 1. So I didn't wrote the expansion upto K terms. (14 Jul '15, 14:20) avmnusng5★
 1 I think setter's solution is wrong on the following test case: 1 2003 2003 1 1 Here N=K=2003, F=L=1, the answer is N/K=1, so P=Q=1, and P*Q^-1 (mod 2003) == 1. The setter's solution outputs 0. answered 13 Jul '15, 21:43 21●1●5 accept rate: 0% Here K is multiple of mod (2003), so Inverse doesn't exist. Answer is correct. (14 Jul '15, 09:02) avmnusng5★ 1 Nope, inverse of Q does exist. K doesn't have anything to do with it. (14 Jul '15, 10:25)
 0 I had a doubt. During the contest I submitted a code for this problem --> Code When I run this code on ideone (actually I used ideone for writing this program), it gives me wrong o/p from my fft function for those values of L which are not powers of 2. For ex, the input:  1 26728 2728 123 4  The o/p I was getting on Ideone was -679. But on CC IDE what I get is the correct answer. I don't know what was the problem behind this. If anyone could help me finding out! Here is my ideone code link: Ideone answered 13 Jul '15, 15:38 1●1●1 accept rate: 0%
 0 I did not understand the part where $F > 1$. How did you arrive at this? Say, if number of such terms is $n_P$, then we add to our answer $\frac{num}{den}$ where $\textrm{num} = \binom{N}{P} * P! * n_P \textrm{ }$ and $den=K^P$ by the same logic as we had in F=1. For F=1 it was simple, the number of ways $A_i$ could have distinct values $= \binom{N}{L} * L! \textrm{ }$ because you could choose L distinct values from N values and permute them to get all possible terms. But in the case where $F > 1$, calculating the number of such terms: $$x_{1, A_1} * x_{1, A_2} * ... * x_{1, A_F} *$$ $$x_{2, B_1} * x_{2, B_2} * ... * x_{2, B_F} *$$ $$...$$ $$x_{L, Z_1} * x_{L, Z_2} * ... * x_{L, Z_F} *$$ such that no $A_i = B_j$ and $A_i = C_j$ and so on, doesn't look as easy. Also, for any given problem, value of $F$ is fixed so $P$ should always be equal to $L*F$ so why are you solving it for all $P$ in the range $L$ to $L*F$ ? answered 14 Jul '15, 03:53 350●1●6●12 accept rate: 6%
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question asked: 13 Jul '15, 08:35

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last updated: 19 Jan '16, 18:02