Binary Search

给定三个正整数 $n,x,pos$，请你求出满足以下条件的数组 $a$ 的个数： * 数组 $a$ 是正整数 1~$n$ 的一种排列，且 $a_{pos}=x$（下标从 $0$ 开始） * 对于数组 $a$ 运行如上代码（见图片，就是二分查找的模板），其返回值为`true`。 请将答案对 $1e9+7$ 取模后输出。 翻译 by vectorwyx

Andrey thinks he is truly a successful developer, but in reality he didn't know about the binary search algorithm until recently. After reading some literature Andrey understood that this algorithm allows to quickly find a certain number $ x $ in an array. For an array $ a $ indexed from zero, and an integer $ x $ the pseudocode of the algorithm is as follows: Note that the elements of the array are indexed from zero, and the division is done in integers (rounding down). Andrey read that the algorithm only works if the array is sorted. However, he found this statement untrue, because there certainly exist unsorted arrays for which the algorithm find $ x $ ! Andrey wants to write a letter to the book authors, but before doing that he must consider the permutations of size $ n $ such that the algorithm finds $ x $ in them. A permutation of size $ n $ is an array consisting of $ n $ distinct integers between $ 1 $ and $ n $ in arbitrary order. Help Andrey and find the number of permutations of size $ n $ which contain $ x $ at position $ pos $ and for which the given implementation of the binary search algorithm finds $ x $ (returns true). As the result may be extremely large, print the remainder of its division by $ 10^9+7 $ .

The only line of input contains integers $ n $ , $ x $ and $ pos $ ( $ 1 \le x \le n \le 1000 $ , $ 0 \le pos \le n - 1 $ ) — the required length of the permutation, the number to search, and the required position of that number, respectively.

Print a single number — the remainder of the division of the number of valid permutations by $ 10^9+7 $ .

4 1 2

6

123 42 24

824071958

All possible permutations in the first test case: $ (2, 3, 1, 4) $ , $ (2, 4, 1, 3) $ , $ (3, 2, 1, 4) $ , $ (3, 4, 1, 2) $ , $ (4, 2, 1, 3) $ , $ (4, 3, 1, 2) $ .