Question

How many circular permutations are there given the numbers on the clock?

Answer 1

Solution

There are two possibilities

1. If we consider the clockwise rotation
2. If we consider the counterclockwise rotation

If we consider both the rotations different from one another, then the total number of permutations is given by $(n-1)!$

In this case $n=12$

Therefore, the total permutations will be

$(12-1)!=11!$

$=11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$

$=39916800$

Now if we consider both the rotations are not different from one another, then the total number of permutations is given by $=\frac{(n-1)!}{2!}$

In this case $n=12$

Therefore, the total permutations will be

$=\frac{(12-1)!}{2!}$

$=\frac{(11)!}{2!}$

$=19958400$