(1): Numbers of ways in which all the letters of the word $\bf{"ALASKA"}$ can be arranged in a circle distinguishing between the clockwise and anticlockwise arrangements, is
(2): Numbers of ways in which all the letters of the word $\bf{"SUNDAY"}$ can be arranged in a circle distinguishing between clockwise and anticlockwise arrangements, is
$\bf{My\; Try::}$ For $\bf{1^{st}}$ one:
If we arrange the letters of the word $\bf{"ALASKA"}$ in a row, then the total no. of ways $\displaystyle = \frac{6!}{3!} = 120$
But I did not understand how can I arrange these words along a circle.
Can anyone give me a detailed explanation?
Thanks.
Best Answer
The six distinct letters of the word SUNDAY can be arranged in $6!$ ways along a line. However, if they are arranged along a circle, the six permutations SUNDAY, UNDAYS, NDAYSU, DAYSUN, AYSUND, YSUNDA obtained by successive $60^\circ$ rotations are indistinguishable.
More generally, the six permutations that can be obtained by rotations of $0^\circ$, $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, and $300^\circ$ are indistinguishable. Hence, there are
$$\frac{6!}{6} = 5!$$
indistinguishable clockwise permutations of the word SUNDAY. By the same reasoning, there are $5!$ anti-clockwise permutations of the word SUNDAY, each of which corresponds to reading a given clockwise permutation in reverse order. Since we are distinguishing between clockwise and anti-clockwise arrangements, we count a given permutation and its reflection separately. Therefore, there are $5!$ circular permutations of the word SUNDAY.
As you observed, there are $$\frac{6!}{3!} = 120$$ distinct arrangements of the six letters of the word ALASKA in a line. Since there are six indistinguishable arrangements produced by rotations around the circle, there are $$\frac{6!}{6 \cdot 3!} = 20$$ distinct clockwise arrangements of the letters of the word ALASKA. Since we are treating reflections as distinguishable, there are $20$ distinguishable circular permutations of the word ALASKA.