Tenzing and Random Operations

2000ms

1048576K

Description:

Yet another random problem.

Tenzing has an array $$$a$$$ of length $$$n$$$ and an integer $$$v$$$.

Tenzing will perform the following operation $$$m$$$ times:

Choose an integer $$$i$$$ such that $$$1 \leq i \leq n$$$ uniformly at random.
For all $$$j$$$ such that $$$i \leq j \leq n$$$, set $$$a_j := a_j + v$$$.

Tenzing wants to know the expected value of $$$\prod_{i=1}^n a_i$$$ after performing the $$$m$$$ operations, modulo $$$10^9+7$$$.

Formally, let $$$M = 10^9+7$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output the integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.

Input:

The first line of input contains three integers $$$n$$$, $$$m$$$ and $$$v$$$ ($$$1\leq n\leq 5000$$$, $$$1\leq m,v\leq 10^9$$$).

The second line of input contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1\leq a_i\leq 10^9$$$).

Output:

Output the expected value of $$$\prod_{i=1}^n a_i$$$ modulo $$$10^9+7$$$.

Sample Input:

2 2 5
2 2

Sample Output:

Sample Input:

5 7 9
9 9 8 2 4

Sample Output:

975544726

Note:

There are three types of $$$a$$$ after performing all the $$$m$$$ operations :

1. $$$a_1=2,a_2=12$$$ with $$$\frac{1}{4}$$$ probability.

2. $$$a_1=a_2=12$$$ with $$$\frac{1}{4}$$$ probability.

3. $$$a_1=7,a_2=12$$$ with $$$\frac{1}{2}$$$ probability.

So the expected value of $$$a_1\cdot a_2$$$ is $$$\frac{1}{4}\cdot (24+144) + \frac{1}{2}\cdot 84=84$$$.

Informação

Provedor Codeforces

Código CF1842G