another perms and combs qs srry. (1 Viewer)
- Thread starter yashbb
- Start date
Assuming you mean the letters are arranged in a circle, place A in 1 way and the other 5 in 5! ways, so there are 5! total arrangements,
With the R opposite the A, again place A in 1 way, then R in 1 way (the only place opposite R) and the remainder in 4! ways, so 4! arrangements with A and R opposite.
 &= \cfrac{\text{Arrangements with A and R opposite}}{\text{Total arrangements}} \\ &= \cfrac{4!}{5!} \\ &= \cfrac{4!}{5 \times 4!} \\ &= \cfrac{1}{5} \end{align*})
That this is the answer can be seen intuitively by considering A being placed first... there are then 5 places remaining, only 1 of which is opposite A, so the chance of R being in that position is 1 in 5.
With the R opposite the A, again place A in 1 way, then R in 1 way (the only place opposite R) and the remainder in 4! ways, so 4! arrangements with A and R opposite.
That this is the answer can be seen intuitively by considering A being placed first... there are then 5 places remaining, only 1 of which is opposite A, so the chance of R being in that position is 1 in 5.