What will be the number of permutations and combinations when m objects are to be taken from a group of n objects, having 'a' and 'b' number similar objects?

Example: Find number of ways of selecting 4 people from a group of 6 children and 4 elders.

I want a general formulae to solve such problems.

  • 2
    $\begingroup$ To be perfectly clear, each child is distinct and each elder is distinct, yes? Then why does it matter that of the people available to choose from some are elders and some are children? There are simply ten distinct people (6+4) and we are choosing four of them for a total number of $\binom{10}{4}$ selections. $\endgroup$ – JMoravitz Apr 20 at 18:49
  • $\begingroup$ Sorry, if my question does not make it clear but they are not distinct. $\endgroup$ – manoj Apr 20 at 19:16
  • $\begingroup$ In that case, it sounds like you are asking for the number of integral solutions to the system of equations $\begin{cases}x_1+x_2+\dots+x_k=m\\0\leq x_1\leq c_1\\0\leq x_2\leq c_2\\\vdots\end{cases}$. This is generally solved via stars-and-bars and inclusion-exclusion. The number $n$ doesn't even matter in most cases. $\endgroup$ – JMoravitz Apr 20 at 19:22

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