Score : 100 points
There are N dishes, numbered 1, 2, \ldots, N. Initially, for each i (1 \leq i \leq N), Dish i has a_i (1 \leq a_i \leq 3) pieces of sushi on it.
Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:
Find the expected number of times the operation is performed before all the pieces of sushi are eaten.
Input is given from Standard Input in the following format:
N a_1 a_2 \ldots a_N
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 10^{-9}.
3 1 1 1
5.5
The expected number of operations before the first piece of sushi is eaten, is 1. After that, the expected number of operations before the second sushi is eaten, is 1.5. After that, the expected number of operations before the third sushi is eaten, is 3. Thus, the expected total number of operations is 1 + 1.5 + 3 = 5.5.
1 3
3
Outputs such as 3.00
, 3.000000003
and 2.999999997
will also be accepted.
2 1 2
4.5
10 1 3 2 3 3 2 3 2 1 3
54.48064457488221