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December 2010

PrefixSet edition (2010-10)

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

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 and D is an integer within the range [0..1,000,000];
- T[i] ≥ D[i] for i=0..(N−1).

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