K. Expected Value

Codeforces

IDCF10212K

Time3000ms

Memory16MB

Difficulty—

English · Original

Formal · Original

Here is a game played with sequence $a_1, \\dots, a_n$. On each turn, the player chooses some position $i < n$ uniformly at random, replaces the element $a_i$ with $a_i -a_{i + 1}$, and then removes the element $a_{i + 1}$ from the sequence. This continues until there is only one element left. What is the expected value of the last element? The first line of input contains a single integer $n$ ($2 <= n <= 4000$). The second line of input contains $n$ integers $a_1, \\dots, a_n$ ($1 <= a_i <= 4000$). If the answer is $frac(P, Q)$ such that $P$ and $Q$ are coprime, output a single integer which is $(P dot.op Q^(-1)) bmod (10^9 + 7)$. It is guaranteed that $Q not equiv 0 pmod 10^9 + 7$. Pay attention to the non-standard memory limit. ## Input The first line of input contains a single integer $n$ ($2 <= n <= 4000$).The second line of input contains $n$ integers $a_1, \\dots, a_n$ ($1 <= a_i <= 4000$). ## Output If the answer is $frac(P, Q)$ such that $P$ and $Q$ are coprime, output a single integer which is $(P dot.op Q^(-1)) bmod (10^9 + 7)$. It is guaranteed that $Q not equiv 0 pmod 10^9 + 7$. [samples] ## Note Pay attention to the non-standard memory limit.

**Definitions** Let $ n \in \mathbb{Z} $ with $ 2 \leq n \leq 4000 $. Let $ A = (a_1, a_2, \dots, a_n) $ be a sequence of integers with $ 1 \leq a_i \leq 4000 $. **Process** At each step, choose uniformly at random an index $ i \in \{1, 2, \dots, n-1\} $, and replace the sequence by: $$ (a_1, \dots, a_{i-1}, a_i - a_{i+1}, a_{i+2}, \dots, a_n) $$ This reduces the sequence length by 1. Repeat until one element remains. **Objective** Compute the expected value $ \mathbb{E} $ of the final remaining element, and output $ P \cdot Q^{-1} \mod (10^9 + 7) $, where $ \mathbb{E} = \frac{P}{Q} $ in lowest terms.

API Response (JSON)

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