In arranging people around a circular table we take into acc

In arranging people around a circular table, we take into account their seats relative to each other, not the actual position of any one person. Show that n people can be arranged around a circular table in (n 1)! ways, and In how many ways can 10 symmetrical keys be arranged on a key ring?

Solution

here the relative position of the persons are condiered rather than their actual position.

so at first a particular person\'s position is fixed. so the 2nd person can have any one place from the remaining n-1 places. the 3rd person can have one place from the remaining n-2 places and so on....at last the last person would have only one seat left.

hence total number of arrangment is (n-1)*(n-2)*(n-3)*......*3*2*1=(n-1)! [proved]

so 10 symmetrical keys can be arranged on a key ring in (10-1)!=9!=362880 ways [answer]