Sushi

AtCoder

IDdp_j

Time2000ms

Memory256MB

Difficulty—

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: * Roll a die that shows the numbers $1, 2, \ldots, N$ with equal probabilities, and let $i$ be the outcome. If there are some pieces of sushi on Dish $i$, 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. ## Constraints * All values in input are integers. * $1 \leq N \leq 300$ * $1 \leq a_i \leq 3$ ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $\ldots$ $a_N$ [samples]

Samples

Input #1

3
1 1 1

Output #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 $1 + 1.5 + 3 = 5.5$.

Input #2

1
3

Output #2

3

Outputs such as `3.00`, `3.000000003` and `2.999999997` will also be accepted.

Input #3

2
1 2

Output #3

4.5

Input #4

10
1 3 2 3 3 2 3 2 1 3

Output #4

54.48064457488221

API Response (JSON)

{
  "problem": {
    "name": "Sushi",
    "description": {
      "content": "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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "dp_j"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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.\nTaro will perform the following operation...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments