Super simple solution using combinatorics
Key Insights\n\nThe problem can be viewed as a way of distributing the values in nums between two arrays while maintaining their monotonic properties.\nThe mini
# Key Insights\n\nThe problem can be viewed as a way of distributing the values in nums between two arrays while maintaining their monotonic properties.\nThe minimum possible value for arr2 at any position constrains the possible distributions.\nOnce we find the minimum possible value for arr2, the problem transforms into a combinatorial question.\n\n# Approach\n\nFirst, we iterate through nums to find the minimum possible value for arr2 (min_arr2).\n\nWe maintain two variables, a and b, representing the current values in arr1 and arr2 respectively.\nWe update these values ensuring arr1 is non-decreasing and arr2 is non-increasing.\n\nIf min_arr2 is negative, it\'s impossible to create valid monotonic pairs, so we return 0.\nIf min_arr2 is non-negative, we can view this as a stars and bars problem:\n\nWe need to distribute min_arr2 "stars" into len(nums) + 1 "bars" (including the space before the first element and after the last).\nThis is equivalent to choosing len(nums) positions out of (min_arr2 + len(nums)) total positions.\n\n\nWe use the combination formula to calculate this: C(min_arr2 + len(nums), len(nums)).\nFinally, we return the result modulo $10^9 + 7$ as required by the problem.\n\nTime and Space Complexity\n\nTime Complexity: O(n), where n is the length of nums. We iterate through nums once.\nSpace Complexity: O(1), as we only use a constant amount of extra space.\n\n# Code\n```py\nfrom math import comb\nclass Solution:\n    def countOfPairs(self, nums: List[int]) -> int:\n        MOD = 10**9 + 7\n        a, b = 0, nums[0]\n        arr1 = [a]\n        arr2 = [b]\n        for i in nums[1:]:\n            # min a possible\n            a = max(a, i - b)\n            b = i - a\n            arr1.append(a)\n            arr2.append(b)\n        # probablity question\n        # bars and stars\n        # we have to convert 5, 5, 5, 5 to 0, 0, 0, 0 in the end\n        # xi can be {0, 1, 2, 3, 4, 5}\n        # to find the number of times xi appears is the problem\n        # x0 + x1 + x2 + x3 + x4 + x5 = 4\n        \n        # number of bars = 5\n        # number of stars = 4\n        # 9 choose 4 \n        arr_2_min = min(arr2)\n        if arr_2_min < 0:\n            return 0\n        res = comb(arr_2_min + len(arr2), len(arr2))\n        \n        return res % MOD\n```
['Python3']