Actual Real World Probability Problem

When I was doing my maths degree, I often pondered what the actual uses were for many of the things we did....

Now i've come across a probability question as part of my work, but my brain has officially left the building!

It looks like fairly basic probability stuff, which is quite embarrassing for me especially after dealing such gems as stochastic modelling while a student! Here's a broken down version of the question;

"I have 3 different items in a barrel, A, B and C. They are evenly mixed in the barrel (ie: 1/3 prob of picking each) but are not replaced when they are picked, so this probability will change as we take items out.

For simplicity assume there are 1000 of each item in the barrel to start with, though this number will be subject to change, so the solution should be clearly manipulable to change the input values.

* What is the chance of getting 15 and then 16 items the same?

* What is the chance of __not__ getting at least 3A, 1B and 1C, after both 15 and 16 selections from the barrel."

I've managed to produce a spreadsheet with a changeable input for the number of units which calculates the chance of getting 15 (one in 15.4m) and then 16 (one in 46.7m) of all the same item, but that's as far as I've got!

Should part 2 be as easy as I think it looks, ie: nCr and some conditional formulae? I think perhaps I'm confusing myself. Doesn't help that my memory has become full of P&L's and balance sheets.

I'm considering making the assumption that the probability stays at 1/3 for picking the items, as there are so many items in there, which will hopefully make solving it a lot easier.

Solutions / detailed reminders would be much appreciated!

Thanks,