CF1151F Sonya and Informatics

A girl named Sonya is studying in the scientific lyceum of the Kingdom of Kremland. The teacher of computer science (Sonya's favorite subject!) invented a task for her. Given an array $ a $ of length $ n $ , consisting only of the numbers $ 0 $ and $ 1 $ , and the number $ k $ . Exactly $ k $ times the following happens: - Two numbers $ i $ and $ j $ are chosen equiprobable such that ( $ 1 \leq i < j \leq n $ ). - The numbers in the $ i $ and $ j $ positions are swapped. Sonya's task is to find the probability that after all the operations are completed, the $ a $ array will be sorted in non-decreasing order. She turned to you for help. Help Sonya solve this problem. It can be shown that the desired probability is either $ 0 $ or it can be represented as $ \dfrac{P}{Q} $ , where $ P $ and $ Q $ are coprime integers and $ Q \not\equiv 0~\pmod {10^9+7} $ .

N/A

N/A

In the first example, all possible variants of the final array $ a $ , after applying exactly two operations: $ (0, 1, 0) $ , $ (0, 0, 1) $ , $ (1, 0, 0) $ , $ (1, 0, 0) $ , $ (0, 1, 0) $ , $ (0, 0, 1) $ , $ (0, 0, 1) $ , $ (1, 0, 0) $ , $ (0, 1, 0) $ . Therefore, the answer is $ \dfrac{3}{9}=\dfrac{1}{3} $ . In the second example, the array will not be sorted in non-decreasing order after one operation, therefore the answer is $ 0 $ .