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#### SpaceCrews

Find the number of different ways in which the space crew can be selected.

N countries (numbered from 0 to N−1) participate in a space mission. Each country has trained a certain number of astronauts and each country has to delegate a certain number of astronauts to the mission's crew. How many different ways are there to select the crew?

For example, suppose there are three countries A-land, B-land and C-land and

• A-land has 6 astronauts;
• B-land has 4 astronauts;
• C-land has 7 astronauts.

and

• A-land has to delegate 1 astronaut;
• B-land has to delegate 3 astronauts;
• C-land has to delegate 4 astronauts.

Then

• there are 6 different ways in which A-land can delegate 1 out of 6 astronauts;
• there are 4 different ways in which B-land can delegate 3 out of 4 astronauts;
• there are 35 different ways in which C-land can delegate 4 out of 7 astronauts.

Each country's choice is independent, so the total number of different ways to build the mission crew is 6*4*35=840.

Write a function

class Solution { public int solution(int[] T, int[] D); }

that, given two non-empty arrays T and D consisting of N integers each, returns the number of different ways in which the space crew can be selected, where:

• T[K] = number of astronauts in country K;
• D[K] = number of astronauts to be delegated from country K.

For example, given N=3 and

T = 6 T = 4 T = 7 D = 1 D = 3 D = 4

the function should return 840, as explained above. If the result exceeds 1,410,000,016, the function should return the remainder of the result modulo 1,410,000,017.

Write an efficient algorithm for the following assumptions:

• N is an integer within the range [1..1,000];
• each element of arrays T, D is an integer within the range [0..1,000,000];
• T[i] ≥ D[i] for i=0..(N−1).