{"problem":{"name":"No Streak","description":{"content":"Alice and Bob played rock-paper-scissors $N$ times. Each round of rock-paper-scissors results in “Alice’s win,” “Bob’s win,” or “Draw.”   Among all possible sequences of results of $N$ rounds, how man","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":3000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"agc070_c"},"statements":[{"statement_type":"Markdown","content":"Alice and Bob played rock-paper-scissors $N$ times. Each round of rock-paper-scissors results in “Alice’s win,” “Bob’s win,” or “Draw.”  \nAmong all possible sequences of results of $N$ rounds, how many satisfy the following conditions? Find the count modulo $\\bf{1000000007 = 10^9 + 7}$.\n\n*   Among the $N$ rounds, Alice won exactly $A$ times and Bob won exactly $B$ times.\n*   Alice never won twice in a row without intervening draws.\n*   Bob never won twice in a row without intervening draws.\n*   At no point did Bob’s number of wins exceed Alice’s number of wins. Formally, for all $1 \\leq i \\leq N$, “the number of times Alice has won up through round $i$” is at least “the number of times Bob has won up through round $i$.”\n\n## Constraints\n\n*   $2 \\leq N \\leq 2 \\times 10^7$\n*   $1 \\leq B \\leq A$\n*   $A + B \\leq N$\n*   $N$, $A$, and $B$ are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$ $A$ $B$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"agc070_c","tags":[],"sample_group":[["5 2 2","5\n\nFor example, the following sequence of $N$ rounds satisfies the conditions:\n\n*   Round $1$: Alice wins.\n*   Round $2$: Bob wins.\n*   Round $3$: Alice wins.\n*   Round $4$: Draw.\n*   Round $5$: Bob wins.\n\nOn the other hand, the following sequence of $N$ rounds does **not** satisfy the conditions, because after round $4$, Alice’s number of wins $(=1)$ is less than Bob’s number of wins $(=2)$:\n\n*   Round $1$: Alice wins.\n*   Round $2$: Bob wins.\n*   Round $3$: Draw.\n*   Round $4$: Bob wins.\n*   Round $5$: Alice wins.\n\nThere are five sequences in total that satisfy the conditions."],["70 29 12","693209192\n\nRemember to find the count modulo $(10^9 + 7)$."],["20000000 1234567 890123","566226457"]],"created_at":"2026-03-03 11:01:14"}}