A geometric interpretation (2 lines in Python)
Let us think of the pairs (arr1[i], arr2[i]) as of the (x,y)-coordinates of a point on a plane. Then each condition of the form arr1[i]+arr2[i]=nums[i] translat
Let us think of the pairs (arr1[i], arr2[i]) as of the (x,y)-coordinates of a point on a plane. Then each condition of the form ```arr1[i]+arr2[i]=nums[i]``` translates to an equation of a line x+y=nums[i]. A monotonic pair can now be interpreted as a sequence of ```n=len(nums)``` points with integer coordinates lying in the first quadrant on the given lines. Futhermore, by monotonicity conditions on ```arr1, arr2```, each point, starting with the second one, should be lower and to the right of the previous one (or coincide with it, if two lines overlap).\n![image](https://assets.leetcode.com/users/images/312ce575-6449-49a4-9c59-a14f7b55cdd4_1723406176.5814037.png)\n\nThus the problem now amounts to computing the number of ways to pick ```n``` integer points  (possibly, with repetitions) on the interval of a certain length (see the picture).\n\n```class Solution:\n    def countOfPairs(self, nums: List[int]) -> int:        \n        total = nums[0] - sum(max(prev - cur, 0) for prev, cur in itertools.pairwise(nums))        \n        return math.comb(len(nums) + total, total) % (10 ** 9 + 7) if total >= 0 else 0\n```
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