# Probability of Drawing Enough Numbers (Combinatorics)

Maybe you could help me with the following problem.

Given a series of incremental numbers that is split in two, so $$s = 1, 2, 3, ..., n_1$$, $$n_1 + 1, n_1 +2 ,..., n_2$$. Also given a integer number $$0 < x < n_2$$.

Now take a random combination of $$x$$ numbers out of $$s$$ (so without replacement and order does not matter).

Two questions:

• What is the probability that the random combination contains all numbers between and including $$1$$ and $$n_1$$? The combination might include numbers larger than $$n_1$$.
• What is the probability that the random combination contains all numbers between and including $$n_1 + 1$$ and $$n_2$$? The combination might include numbers smaller than or equal to $$n_1$$.

Both probabilities would be ideally expressed in terms of $$n_1, n_2$$ and $$x$$.

• How many ways of choosing $$x$$ numbers without replacement from $$n_2$$ numbers
• How many ways of choosing $$n_1$$ numbers without replacement from $$n_1$$ numbers and $$x-n_1$$ numbers without replacement from $$n_2-n_1$$ other numbers
• The answer to your second question is the same as the answer to your first but replacing $$n_1$$ by $$n_2-n_1$$