Best-of-(2n-1)

AtCoder

IDm_solutions2019_c

Time2000ms

Memory256MB

Difficulty—

Takahashi and Aoki will play a game. They will repeatedly play it until one of them have $N$ wins in total. When they play the game once, Takahashi wins with probability $A$ %, Aoki wins with probability $B$ %, and the game ends in a draw (that is, nobody wins) with probability $C$ %. Find the expected number of games that will be played, and print it as follows. We can represent the expected value as $P/Q$ with coprime integers $P$ and $Q$. Print the integer $R$ between $0$ and $10^9+6$ (inclusive) such that $R \times Q \equiv P\pmod {10^9+7}$. (Such an integer $R$ always uniquely exists under the constraints of this problem.) ## Constraints * $1 \leq N \leq 100000$ * $0 \leq A,B,C \leq 100$ * $1 \leq A+B$ * $A+B+C=100$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A$ $B$ $C$ [samples]

Samples

Input #1

1 25 25 50

Output #1

2

Since $N=1$, they will repeat the game until one of them wins. The expected number of games played is $2$.

Input #2

4 50 50 0

Output #2

312500008

$C$ may be $0$.

Input #3

1 100 0 0

Output #3

1

$B$ may also be $0$.

Input #4

100000 31 41 28

Output #4

104136146

API Response (JSON)

{
  "problem": {
    "name": "Best-of-(2n-1)",
    "description": {
      "content": "Takahashi and Aoki will play a game. They will repeatedly play it until one of them have $N$ wins in total. When they play the game once, Takahashi wins with probability $A$ %, Aoki wins with probabil",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "m_solutions2019_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi and Aoki will play a game. They will repeatedly play it until one of them have $N$ wins in total.\nWhen they play the game once, Takahashi wins with probability $A$ %, Aoki wins with probabil...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments