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A small kingdom wants to choose their king, his advisor and 5 guards, how many ways are there to choose them? $30 \choose 7$? Why? Why not?

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Not, because it is not specified who is who. The right answer is $${30\choose 1}\cdot {29\choose 1}\cdot {28\choose 5}$$ That is you first choose the king among 30 people and then advisor among other 29 and finaly 5 guards among 28 people.


But you can repair your solution: $$ {30\choose 7}\cdot {7\choose 1}\cdot {6\choose 1}$$ That is you first choose a 7 people among 30 people and then king among choosen and then advisor among other 6 you choose and finaly you are left with 5 people which are to be guards.

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  • $\begingroup$ I still don't understand. You say:"Because it is not specified who is who", what does that mean? $\endgroup$
    – Coder-Man
    Oct 11, 2018 at 19:57
  • $\begingroup$ When you choose 7 people we dont know who is king among themand so on... $\endgroup$
    – nonuser
    Oct 11, 2018 at 19:59
  • $\begingroup$ But why do we care who is king? We just choose 7 people to work in "the government". $\endgroup$
    – Coder-Man
    Oct 11, 2018 at 19:59
  • $\begingroup$ Well, then you don't need king, advisor and guard... $\endgroup$
    – nonuser
    Oct 11, 2018 at 20:01

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