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We have $25$ countries and $100$ teams. Teams can have variable sizes. Each team consists of a combination of players from different countries. Now we have to select $13$ teams in total subjected to following condition:

  • If $i^{th}$ team is selected which has a player from $j^{th}$ country, then no teams having players of $j^{th}$ country can be selected.
  • If say for example ,$3^{rd}$ team is selected with $2$ players $(P_0, P_1)$ in country $1$ and $1$ player in country $2$. So in one selection I can take $P_0$ and in other $P_1$ leading to 2 ways.

$T[i][j]$ represents the number of players of country $j$ in Team $i$.

For example

  • $T[3][1] = 2$ means $3^{rd}$ team has $2$ players from country $1$.
  • $T[1][2] = 7$ means $1^{st}$ team has $7$ players from country $2$.

We are given the matrix $T$. In how many ways can we select $13$ teams?

If question is not clear, it is basically the number of ways of selecting $13$ rows in a $25X100$ matrix such that the resulting $13X100$ matrix has columns with all values $0$ except at 1 place.

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  • $\begingroup$ How are the values in $T$ relevant? Isn't all that's relevant whether they're non-zero? $\endgroup$ – joriki May 8 '16 at 13:17
  • $\begingroup$ Lets say if I select team $3$ with $2$ players $(P_0, P_1)$ in country $1$ and $1$ player in country $2$. So in one selection I can take $P_0$ and in other $P_1$ leading to 2 ways. I will add it in the EDIT history. $\endgroup$ – maverick May 8 '16 at 13:33
  • $\begingroup$ The question has now become very unclear. You should differentiate clearly between a "team" in the sense of the $100$ pre-existing teams with given numbers of players and what you are now also calling a "team" but should be referred to by a different term (e.g. "selection"), which apparently has at most one player from any given country. $\endgroup$ – joriki May 8 '16 at 14:26

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