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?
2026-03-28 12:33:36.1774701216
Combinations of 7 people out of 30
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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.