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How do I generate a random number between 0 and some integer n

First you would need to create an instance of Random class:
   Random random = new Random();

Then, you can get your random number by calling:
   int pick = random.nextInt(maxvalue);

This would return a value from 0 (inclusive) to maxvalue (exclusive).

For a more cryptographically strong pseudo random generator you may check out java.security.SecureRandom class. Here the caller may specify the algorithm name and (optionally) the package provider. Here is a sample code of using SecureRandom class:
   SecureRandom random = SecureRandom.getInstance("SHA1PRNG");
   random.setSeed(seed);
   int randInt = random.nextInt(maxvalue);


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