White and Black Balls

AtCoder

IDabc205_e

Time2000ms

Memory256MB

Difficulty—

How many ways are there to arrange $N$ white balls and $M$ black balls in a row from left to right to satisfy the following condition? * For each $i$ $(1 \leq i \leq N + M)$, let $w_i$ and $b_i$ be the number of white balls and black balls among the leftmost $i$ balls, respectively. Then, $w_i \leq b_i + K$ holds for every $i$. Since the count can be enormous, find it modulo $(10^9 + 7)$. ## Constraints * $0 \leq N, M \leq 10^6$ * $1 \leq N + M$ * $0 \leq K \leq N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ [samples]

Samples

Input #1

2 3 1

Output #1

9

There are $10$ ways to arrange $2$ white balls and $3$ black balls in a row, as shown below, where `w` and `b` stand for a white ball and a black ball, respectively.
`wwbbb` `wbwbb` `wbbwb` `wbbbw` `bwwbb` `bwbwb` `bwbbw` `bbwwb` `bbwbw` `bbbww`
Among them, `wwbbb` is the only one that does not satisfy the condition. Here, there are $2$ white balls and $0$ black balls among the $2$ leftmost balls, and we have $2 > 0 + K = 1$.

Input #2

1 0 0

Output #2

0

There may be no way to satisfy the condition.

Input #3

1000000 1000000 1000000

Output #3

192151600

Be sure to print the count modulo $(10^9 + 7)$.

API Response (JSON)

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      "content": "How many ways are there to arrange $N$ white balls and $M$ black balls in a row from left to right to satisfy the following condition? *   For each $i$ $(1 \\leq i \\leq N + M)$, let $w_i$ and $b_i$ be",
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      "statement_type": "Markdown",
      "content": "How many ways are there to arrange $N$ white balls and $M$ black balls in a row from left to right to satisfy the following condition?\n\n*   For each $i$ $(1 \\leq i \\leq N + M)$, let $w_i$ and $b_i$ be...",
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