These problems are from the atcoder DP contest, and were transferred onto DMOJ. All problem statements were made by several atcoder users. As there is no access to the test data, all data is randomly generated. If there are issues with the statement or data, please contact or on slack.
There are dishes, numbered . Initially, for each , Dish has pieces of sushi on it.
Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:
- Roll a die that shows the numbers with equal probabilities, and let be the outcome. If there are some pieces of sushi on Dish , eat one of them; if there is none, do nothing.
Find the expected number of times the operation is performed before all the pieces of sushi are eaten.
- All values in input are integers.
The first line will contain the integer .
The next line will contain integers, .
Print the expected number of times the operation is performed before all the pieces of sushi are eaten. The output is considered correct when the relative difference is not greater than .
Sample Input 1
3 1 1 1
Sample Output 1
Explanation For Sample 1
The expected number of operations before the first piece of sushi is eaten, is . After that, the expected number of operations before the second sushi is eaten, is . After that, the expected number of operations before the third sushi is eaten, is . Thus, the expected total number of operations is .
Sample Input 2
Sample Output 2
Explanation For Sample 2
Outputs such as
2.999999997 will also be accepted.
Sample Input 3
2 1 2
Sample Output 3
Sample Input 4
10 1 3 2 3 3 2 3 2 1 3
Sample Output 4