{"problem":{"name":"White and Black Balls","description":{"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","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc205_e"},"statements":[{"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 the number of white balls and black balls among the leftmost $i$ balls, respectively. Then, $w_i \\leq b_i + K$ holds for every $i$.\n\nSince the count can be enormous, find it modulo $(10^9 + 7)$.\n\n## Constraints\n\n*   $0 \\leq N, M \\leq 10^6$\n*   $1 \\leq N + M$\n*   $0 \\leq K \\leq N$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$ $K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc205_e","tags":[],"sample_group":[["2 3 1","9\n\nThere 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.\n`wwbbb` `wbwbb` `wbbwb` `wbbbw` `bwwbb` `bwbwb` `bwbbw` `bbwwb` `bbwbw` `bbbww`\nAmong 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 $2 > 0 + K = 1$."],["1 0 0","0\n\nThere may be no way to satisfy the condition."],["1000000 1000000 1000000","192151600\n\nBe sure to print the count modulo $(10^9 + 7)$."]],"created_at":"2026-03-03 11:01:14"}}