{"problem":{"name":"Random Max","description":{"content":"There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval $[L_i, R_i]$. (That is, $x_i$ is a random variable that can tak","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":3000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"hhkb2020_f"},"statements":[{"statement_type":"Markdown","content":"There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval $[L_i, R_i]$. (That is, $x_i$ is a random variable that can take any real value between $L_i$ and $R_i$ (inclusive) with equal probability.)\nLet $E$ be the expected value of the maximum of these $N$ random variables. Under the constraints in this problem, it can be proved that $E \\times (N+1)! \\times \\prod_{i=1}^N (R_i - L_i)$ is a positive integer. Find this value modulo $1,000,000,007$.\n\n## Constraints\n\n*   $1 ≤ N ≤ 1000$\n*   $0 ≤ L_i < R_i ≤ 10^9$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$L_1$ $R_1$\n$:$\n$L_N$ $R_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"hhkb2020_f","tags":[],"sample_group":[["1\n1 2","3\n\nThe expected value of the maximum of these random variables - there is actually just one in this case - is equal to the median of the range of the values the variable can take, that is, $E = \\frac{3}{2}$.\nThus, the correct output is $E \\times (N+1)! \\times (R_1 - L_1) = E \\times 2 = 3$."],["2\n1 2\n1 2","10\n\nThe expected value in question is $E = \\frac{5}{3}$."],["2\n1 2\n2 4","36"],["5\n40 96\n81 92\n16 384\n32 768\n65 536","52776507"]],"created_at":"2026-03-03 11:01:14"}}