Score : 900 points
You are developing frog-shaped robots, and decided to race them against each other.
First, you placed N robots onto a number line. These robots are numbered 1 through N. The current coordinate of robot i is x_i. Here, all x_i are integers, and 0 < x_1 < x_2 < ... < x_N.
You will repeatedly perform the following operation:
When the coordinate of a robot becomes 0 or less, the robot is considered finished and will be removed from the number line immediately. You will repeat the operation until all the robots finish the race.
Depending on your choice in the operation, the N robots can finish the race in different orders. In how many different orders can the N robots finish the race? Find the answer modulo 10^9+7.
The input is given from Standard Input in the following format:
N x_1 x_2 ... x_N
Print the number of the different orders in which the N robots can finish the race, modulo 10^9+7.
3 1 2 3
4
There are four different orders in which the three robots can finish the race:
3 2 3 4
6
There are six different orders in which the three robots can finish the race:
For example, the order (Robot 3 → Robot 2 → Robot 1) can be achieved as shown in the figure below:
8 1 2 3 5 7 11 13 17
10080
13 4 6 8 9 10 12 14 15 16 18 20 21 22
311014372
Remember to print the answer modulo 10^9+7. This case is not included in the test set for the partial score.