increment of coins

AtCoder

IDabc184_d

Time2000ms

Memory256MB

Difficulty—

We have a bag containing $A$ gold coins, $B$ silver coins, and $C$ bronze coins. Until the bag contains $100$ coins of the same color, we will repeat the following operation: Operation: Randomly take out one coin from the bag. (Every coin has an equal probability of being chosen.) Then, put back into the bag two coins of the same kind as the removed coin. Find the expected value of the number of times the operation is done. ## Constraints * $0 \leq A,B,C \leq 99$ * $A+B+C \geq 1$ ## Input Input is given from Standard Input in the following format: $A$ $B$ $C$ [samples]

Samples

Input #1

99 99 99

Output #1

1.000000000

No matter what coin we take out in the first operation, the bag will contain $100$ coins of that kind.

Input #2

98 99 99

Output #2

1.331081081

We will do the second operation only if we take out a gold coin in the first operation. Thus, the expected number of operations is $2\times \frac{98}{98+99+99}+1\times \frac{99}{98+99+99}+1\times \frac{99}{98+99+99}=1.331081081\ldots$

Input #3

0 0 1

Output #3

99.000000000

Each operation adds a bronze coin.

Input #4

31 41 59

Output #4

91.835008202

API Response (JSON)

{
  "problem": {
    "name": "increment of coins",
    "description": {
      "content": "We have a bag containing $A$ gold coins, $B$ silver coins, and $C$ bronze coins. Until the bag contains $100$ coins of the same color, we will repeat the following operation: Operation: Randomly take ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc184_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a bag containing $A$ gold coins, $B$ silver coins, and $C$ bronze coins.\nUntil the bag contains $100$ coins of the same color, we will repeat the following operation:\nOperation: Randomly take ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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