CF1845E Boxes and Balls

There are $ n $ boxes placed in a line. The boxes are numbered from $ 1 $ to $ n $ . Some boxes contain one ball inside of them, the rest are empty. At least one box contains a ball and at least one box is empty. In one move, you have to choose a box with a ball inside and an adjacent empty box and move the ball from one box into another. Boxes $ i $ and $ i+1 $ for all $ i $ from $ 1 $ to $ n-1 $ are considered adjacent to each other. Boxes $ 1 $ and $ n $ are not adjacent. How many different arrangements of balls exist after exactly $ k $ moves are performed? Two arrangements are considered different if there is at least one such box that it contains a ball in one of them and doesn't contain a ball in the other one. Since the answer might be pretty large, print its remainder modulo $ 10^9+7 $ .

N/A

N/A

In the first example, there are the following possible arrangements: - 0 1 1 0 — obtained after moving the ball from box $ 1 $ to box $ 2 $ ; - 1 0 0 1 — obtained after moving the ball from box $ 3 $ to box $ 4 $ ; - 1 1 0 0 — obtained after moving the ball from box $ 3 $ to box $ 2 $ . In the second example, there are the following possible arrangements: - 1 0 1 0 — three ways to obtain that: just reverse the operation performed during the first move; - 0 1 0 1 — obtained from either of the first two arrangements after the first move.