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문제

수학 선생님 Albert는 사탕을 좋아하는 반 아이들을 위해 재미있는 놀이를 고안했다. 아이들은 아직 어려서 자연수만 알고 있고, 각 아이마다 알고 있는 자연수의 범위가 다르다. 편의상 아이들은 1부터 n까지 번호가 붙어있다고 하고, i번째 아이는 1이상 x[i] 이하의 자연수를 알고 있다고 가정하자. 

놀이는 아래와 같은 규칙에 따라 진행된다.

  1. 각 턴이 시작되면 아이들은 각자 알고 있는 자연수 중 아무 수나 골라서 종이에 적는다.
  2. 모두 원하는 수를 적었으면, Albert 선생님이 n장의 종이를 모은다.
  3. Albert 선생님은 이 수들을 비내림차순으로 정렬하여 길이 n인 수열을 만든다 (수열 A라고 하자).
  4. 만약 수열 A가 이전 턴에 한 번이라도 만들어진 적이 있으면 수열은 버려지고 게임은 계속 진행된다. 
  5. 만약 수열 A가 이전 턴에 만들어진 적이 없다면, Albert선생님이 아이들에게 사탕을 하나씩 나누어주고 게임이 계속 진행된다.
  6. 모든 가능한 (정렬된) 수열이 만들어진 후에 게임이 완전히 끝나게 된다.

모든 아이들이 자연수의 일부만 알고 있기 때문에 이 게임은 언젠가 끝나게 된다. 하지만 Albert 선생님은 미리 사탕을 준비해야 하므로 최대 몇 개를 나눠줘야 할지 알고 싶다. 학생 수 n과 각 학생이 알고 있는 자연수의 범위가 주어졌을 때, Albert 선생님이 나눠줘야 하는 사탕의 최대 수를 구해서 선생님을 도와주도록 하자. 다만 이 수는 매우 클 수 있으므로 1,000,000,007로 나눈 나머지를 구해주기로 했다.

예를 들어 Albert선생님의 반에 두 명의 아이가 있고 이 두 아이는 각자 1이상 3이하의 자연수만 안다고 해보자. 이 경우 총 여섯 종류의 (비내림차순으로) 정렬된 수열을 얻을 수 있다: {1, 1}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, {3, 3}. 각 수열에 대해 처음으로 만들어지는 턴에 Albert 선생님이 사탕 두 개를 나누어 주어야 하므로 총 12개의 사탕을 미리 준비해놔야 한다.

입력

첫 줄에 테스트 케이스의 수 T가 주어진다 (1 <= T <= 10).

각 테스트 케이스는 두 줄에 걸쳐서 주어진다. 첫 줄에 학생 수 n이 주어지고 (1 <= n <= 200) 둘째 줄에 배열 x[]를 표현하는  n개의 자연수가 공백으로 구분되어 주어진다. 각 i에 대해 1 <= x[i] <= 200을 만족한다.

출력

각 테스트 케이스에 대해 Albert선생님이 준비해야 하는 사탕의 최대 수를 1,000,000,007로 나눈 나머지를 출력한다. 

예제 입력 1

4
1
3
2
3 3
3
1 2 3
6
95 96 97 98 99 100

예제 출력 1

3
12
15
627383157

힌트

  • 케이스 1: 아이 한 명이 1이상 3이하의 수를 알기 때문에 정답은 3이다. 정렬된 수열은 {1}, {2}, 그리고 {3} 이렇게 세 가지 존재한다.
  • 케이스 2: 문제에서 예시로 사용되었다.
  • 케이스 3: 이 경우 총 다섯 개의 정렬된 수열이 존재한다: {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 2, 2}, {1, 2, 3}. 학생이 세 명 있기 때문에 이 수열 각각이 처음으로 만들어지는 턴에 Albert는 사탕을 3개씩 나눠주어야 하므로 총 15개의 사탕을 준비해야 한다.
  • 케이스 4: 답이 매우 커질 수 있으므로 1,000,000,007로 나눈 나머지를 출력해야 함에 유의하자. 이 경우 총 1,604,563,870 가지의 정렬된 수열이 존재하고 여섯 명의 아이가 있으므로 사탕은 총 9,627,383,220 개 필요하다. 9,627,383,220 를 1,000,000,007 으로 나눈 나머지는 627,383,157이다.
[{"problem_id":"17137","problem_lang":"0","title":"\uc0ac\ud0d5 \ub180\uc774","description":"<p>\uc218\ud559 \uc120\uc0dd\ub2d8 Albert\ub294 \uc0ac\ud0d5\uc744 \uc88b\uc544\ud558\ub294 \ubc18 \uc544\uc774\ub4e4\uc744 \uc704\ud574 \uc7ac\ubbf8\uc788\ub294 \ub180\uc774\ub97c \uace0\uc548\ud588\ub2e4. \uc544\uc774\ub4e4\uc740 \uc544\uc9c1 \uc5b4\ub824\uc11c \uc790\uc5f0\uc218\ub9cc \uc54c\uace0 \uc788\uace0, \uac01 \uc544\uc774\ub9c8\ub2e4 \uc54c\uace0 \uc788\ub294 \uc790\uc5f0\uc218\uc758 \ubc94\uc704\uac00 \ub2e4\ub974\ub2e4. \ud3b8\uc758\uc0c1 \uc544\uc774\ub4e4\uc740 1\ubd80\ud130 n\uae4c\uc9c0 \ubc88\ud638\uac00 \ubd99\uc5b4\uc788\ub2e4\uace0 \ud558\uace0, i\ubc88\uc9f8 \uc544\uc774\ub294 1\uc774\uc0c1 x[i] \uc774\ud558\uc758 \uc790\uc5f0\uc218\ub97c \uc54c\uace0 \uc788\ub2e4\uace0 \uac00\uc815\ud558\uc790.&nbsp;<\/p>\r\n\r\n<p>\ub180\uc774\ub294 \uc544\ub798\uc640 \uac19\uc740 \uaddc\uce59\uc5d0 \ub530\ub77c \uc9c4\ud589\ub41c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\uac01 \ud134\uc774 \uc2dc\uc791\ub418\uba74 \uc544\uc774\ub4e4\uc740 \uac01\uc790 \uc54c\uace0 \uc788\ub294 \uc790\uc5f0\uc218 \uc911 \uc544\ubb34 \uc218\ub098 \uace8\ub77c\uc11c \uc885\uc774\uc5d0 \uc801\ub294\ub2e4.<\/li>\r\n\t<li>\ubaa8\ub450 \uc6d0\ud558\ub294 \uc218\ub97c \uc801\uc5c8\uc73c\uba74, Albert \uc120\uc0dd\ub2d8\uc774 n\uc7a5\uc758 \uc885\uc774\ub97c \ubaa8\uc740\ub2e4.<\/li>\r\n\t<li>Albert \uc120\uc0dd\ub2d8\uc740 \uc774 \uc218\ub4e4\uc744 \ube44\ub0b4\ub9bc\ucc28\uc21c\uc73c\ub85c \uc815\ub82c\ud558\uc5ec \uae38\uc774 n\uc778 \uc218\uc5f4\uc744 \ub9cc\ub4e0\ub2e4 (\uc218\uc5f4 A\ub77c\uace0 \ud558\uc790).<\/li>\r\n\t<li>\ub9cc\uc57d \uc218\uc5f4 A\uac00 \uc774\uc804 \ud134\uc5d0 \ud55c \ubc88\uc774\ub77c\ub3c4 \ub9cc\ub4e4\uc5b4\uc9c4 \uc801\uc774 \uc788\uc73c\uba74 \uc218\uc5f4\uc740 \ubc84\ub824\uc9c0\uace0 \uac8c\uc784\uc740 \uacc4\uc18d \uc9c4\ud589\ub41c\ub2e4.&nbsp;<\/li>\r\n\t<li>\ub9cc\uc57d \uc218\uc5f4 A\uac00 \uc774\uc804 \ud134\uc5d0 \ub9cc\ub4e4\uc5b4\uc9c4 \uc801\uc774 \uc5c6\ub2e4\uba74, Albert\uc120\uc0dd\ub2d8\uc774 \uc544\uc774\ub4e4\uc5d0\uac8c \uc0ac\ud0d5\uc744 \ud558\ub098\uc529 \ub098\ub204\uc5b4\uc8fc\uace0 \uac8c\uc784\uc774 \uacc4\uc18d \uc9c4\ud589\ub41c\ub2e4.<\/li>\r\n\t<li>\ubaa8\ub4e0 \uac00\ub2a5\ud55c (\uc815\ub82c\ub41c) \uc218\uc5f4\uc774 \ub9cc\ub4e4\uc5b4\uc9c4 \ud6c4\uc5d0 \uac8c\uc784\uc774 \uc644\uc804\ud788 \ub05d\ub098\uac8c \ub41c\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\ubaa8\ub4e0 \uc544\uc774\ub4e4\uc774 \uc790\uc5f0\uc218\uc758 \uc77c\ubd80\ub9cc \uc54c\uace0 \uc788\uae30 \ub54c\ubb38\uc5d0 \uc774 \uac8c\uc784\uc740 \uc5b8\uc820\uac00 \ub05d\ub098\uac8c \ub41c\ub2e4. \ud558\uc9c0\ub9cc Albert \uc120\uc0dd\ub2d8\uc740 \ubbf8\ub9ac \uc0ac\ud0d5\uc744 \uc900\ube44\ud574\uc57c \ud558\ubbc0\ub85c \ucd5c\ub300 \uba87 \uac1c\ub97c \ub098\ub220\uc918\uc57c \ud560\uc9c0 \uc54c\uace0 \uc2f6\ub2e4. \ud559\uc0dd \uc218 n\uacfc \uac01 \ud559\uc0dd\uc774 \uc54c\uace0 \uc788\ub294 \uc790\uc5f0\uc218\uc758 \ubc94\uc704\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, Albert \uc120\uc0dd\ub2d8\uc774 \ub098\ub220\uc918\uc57c \ud558\ub294 \uc0ac\ud0d5\uc758 \ucd5c\ub300 \uc218\ub97c \uad6c\ud574\uc11c \uc120\uc0dd\ub2d8\uc744 \ub3c4\uc640\uc8fc\ub3c4\ub85d \ud558\uc790. \ub2e4\ub9cc \uc774 \uc218\ub294 \ub9e4\uc6b0 \ud074 \uc218 \uc788\uc73c\ubbc0\ub85c 1,000,000,007\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \uad6c\ud574\uc8fc\uae30\ub85c \ud588\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4 Albert\uc120\uc0dd\ub2d8\uc758 \ubc18\uc5d0 \ub450 \uba85\uc758 \uc544\uc774\uac00 \uc788\uace0 \uc774 \ub450 \uc544\uc774\ub294 \uac01\uc790 1\uc774\uc0c1 3\uc774\ud558\uc758 \uc790\uc5f0\uc218\ub9cc \uc548\ub2e4\uace0 \ud574\ubcf4\uc790. \uc774 \uacbd\uc6b0 \ucd1d \uc5ec\uc12f \uc885\ub958\uc758 (\ube44\ub0b4\ub9bc\ucc28\uc21c\uc73c\ub85c) \uc815\ub82c\ub41c \uc218\uc5f4\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4: {1, 1}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, {3, 3}. \uac01 \uc218\uc5f4\uc5d0 \ub300\ud574 \ucc98\uc74c\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \ud134\uc5d0 Albert \uc120\uc0dd\ub2d8\uc774 \uc0ac\ud0d5 \ub450 \uac1c\ub97c \ub098\ub204\uc5b4 \uc8fc\uc5b4\uc57c \ud558\ubbc0\ub85c \ucd1d 12\uac1c\uc758 \uc0ac\ud0d5\uc744 \ubbf8\ub9ac \uc900\ube44\ud574\ub194\uc57c \ud55c\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4 (1 &lt;= T &lt;= 10).<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ub450 \uc904\uc5d0 \uac78\uccd0\uc11c \uc8fc\uc5b4\uc9c4\ub2e4. \uccab \uc904\uc5d0 \ud559\uc0dd \uc218 n\uc774 \uc8fc\uc5b4\uc9c0\uace0 (1 &lt;= n &lt;= 200) \ub458\uc9f8 \uc904\uc5d0 \ubc30\uc5f4 x[]\ub97c \ud45c\ud604\ud558\ub294 &nbsp;n\uac1c\uc758 \uc790\uc5f0\uc218\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 i\uc5d0 \ub300\ud574 1 &lt;= x[i] &lt;= 200\uc744 \ub9cc\uc871\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574 Albert\uc120\uc0dd\ub2d8\uc774 \uc900\ube44\ud574\uc57c \ud558\ub294 \uc0ac\ud0d5\uc758 \ucd5c\ub300 \uc218\ub97c 1,000,000,007\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.&nbsp;<\/p>\r\n","hint":"<ul>\r\n\t<li>\ucf00\uc774\uc2a4 1: \uc544\uc774 \ud55c \uba85\uc774 1\uc774\uc0c1 3\uc774\ud558\uc758 \uc218\ub97c \uc54c\uae30 \ub54c\ubb38\uc5d0 \uc815\ub2f5\uc740 3\uc774\ub2e4. \uc815\ub82c\ub41c \uc218\uc5f4\uc740 {1}, {2}, \uadf8\ub9ac\uace0 {3} \uc774\ub807\uac8c \uc138 \uac00\uc9c0 \uc874\uc7ac\ud55c\ub2e4.<\/li>\r\n\t<li>\ucf00\uc774\uc2a4 2: \ubb38\uc81c\uc5d0\uc11c \uc608\uc2dc\ub85c \uc0ac\uc6a9\ub418\uc5c8\ub2e4.<\/li>\r\n\t<li>\ucf00\uc774\uc2a4 3: \uc774 \uacbd\uc6b0 \ucd1d \ub2e4\uc12f \uac1c\uc758 \uc815\ub82c\ub41c \uc218\uc5f4\uc774 \uc874\uc7ac\ud55c\ub2e4: {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 2, 2}, {1, 2, 3}. \ud559\uc0dd\uc774 \uc138 \uba85 \uc788\uae30 \ub54c\ubb38\uc5d0 \uc774 \uc218\uc5f4 \uac01\uac01\uc774 \ucc98\uc74c\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \ud134\uc5d0 Albert\ub294 \uc0ac\ud0d5\uc744 3\uac1c\uc529 \ub098\ub220\uc8fc\uc5b4\uc57c \ud558\ubbc0\ub85c \ucd1d 15\uac1c\uc758 \uc0ac\ud0d5\uc744 \uc900\ube44\ud574\uc57c \ud55c\ub2e4.<\/li>\r\n\t<li>\ucf00\uc774\uc2a4 4: \ub2f5\uc774 \ub9e4\uc6b0 \ucee4\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c 1,000,000,007\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud574\uc57c \ud568\uc5d0 \uc720\uc758\ud558\uc790. \uc774 \uacbd\uc6b0 \ucd1d 1,604,563,870 \uac00\uc9c0\uc758 \uc815\ub82c\ub41c \uc218\uc5f4\uc774 \uc874\uc7ac\ud558\uace0 \uc5ec\uc12f \uba85\uc758 \uc544\uc774\uac00 \uc788\uc73c\ubbc0\ub85c \uc0ac\ud0d5\uc740 \ucd1d 9,627,383,220 \uac1c \ud544\uc694\ud558\ub2e4. 9,627,383,220 \ub97c 1,000,000,007 \uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub294 627,383,157\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n","original":"1","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"17137","problem_lang":"1","title":"Number Game","description":"<p>Albert is a teacher, and he&rsquo;s teaching n children in his math class. Everyone loves candies, so he&rsquo;s invented a fun game for them to play while he can get their attention by giving out candies.&nbsp;<\/p>\r\n\r\n<p>Here is how the game works. Assume that the children in Albert&rsquo;s class are numbered from 1 to n (there are n children in his class). Child i knows of integers between 1 and x[i], inclusive.&nbsp;<\/p>\r\n\r\n<p>The game plays out for possibly many turns as follows.<\/p>\r\n\r\n<ol>\r\n\t<li>At the beginning of each turn, every child picks an arbitrary number that he\/she knows of, and writes it down on a piece of paper.&nbsp;<\/li>\r\n\t<li>After everyone finishes writing, Albert collects n pieces of paper.<\/li>\r\n\t<li>Albert then sorts the numbers in non-decreasing order, and obtains a sorted list of numbers.<\/li>\r\n\t<li>If this sorted list was obtained in any of the previous turns, the list is discarded and the game continues. If it has not been obtained previously, then every child will get a candy from Albert, and the game continues.<\/li>\r\n\t<li>The game will end when every possible sorted list of numbers has been obtained.<\/li>\r\n<\/ol>\r\n\r\n<p>Because everyone knows of only a finite number of integers, the game will eventually end, but Albert needs to figure out how many candies he needs to prepare in the worst case. Given n (the number of children) and an array x[] (where x[i] is the largest number child i knows of), compute the maximum number of candies Albert will need to give out if the game plays out until every sorted list has been obtained. This number can be quite large, so you only need to compute this number modulo 1,000,000,007.<\/p>\r\n\r\n<p>For instance, suppose that there are two children Alice and Elise in Albert&rsquo;s class and Alice knows 3 integers (1, 2, and 3) and Elise knows 3 integers (1, 2, and 3). In this case, there are six sorted lists of two integers that can be obtained during the game: {1, 1}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, and {3, 3}. Each time one of these lists is obtained for the first time, Albert will need to give out two candies, and therefore Albert will need 12 candies.&nbsp;<\/p>\r\n","input":"<p>The first line will contain T (1 &lt;= T &lt;= 10), the number of test cases.<\/p>\r\n\r\n<p>Each test case will consist of two lines. The first line will contain n (1 &lt;= n &lt;= 200), the number of children in Albert&rsquo;s math class. The second line will contain n integers (delimited by a whitespace) that describe the array x[]. Assume that 1 &lt;= x[i] &lt;= 200 for all i.<\/p>\r\n","output":"<p>For each test case, output the maximum number of candies Albert needs to give out.<\/p>\r\n","hint":"<ul>\r\n\t<li>Case 1: With only one child who knows 3 integers, the answer is trivially 3. The three sorted lists are: {1}, {2}, and {3} and each time one of these lists is obtained for the first time, Albert will need to give out one candy.<\/li>\r\n\t<li>Case 2: This is described in the problem statement.&nbsp;<\/li>\r\n\t<li>Case 3: The number of sorted lists is five: {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 2, 2}, {1, 2, 3}. Since there are three children, Albert will need to prepare 15 candies.<\/li>\r\n\t<li>Case 4: Recall that you will need to compute the answer modulo 1,000,000,007 because the answer can become quite large. There are 1,604,563,870 sorted lists in this case, and there are six children (hence Albert needs to prepare 9,627,383,220 candies). 9,627,383,220 modulo 1,000,000,007 is equal to 627,383,157.<\/li>\r\n<\/ul>\r\n","original":"0","problem_lang_code":"\uc601\uc5b4"}]