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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 $x$.

I hope I expressed the problem clearly, otherwise I can add more information.

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  • $\begingroup$ Welcome to math.SE! Please use MathJax to typeset your math, rather than using backticks. $\endgroup$ – LarrySnyder610 May 24 at 14:52
  • $\begingroup$ Ah, I see you did. :) $\endgroup$ – LarrySnyder610 May 24 at 14:52
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Hints:

  • 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 $n_2-n_1$

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