# How many ways are there to rearrange the letters "SUPERMAN" on a circle with "SUPER" as a continuous substring?

I calculate 48, but apparently this is not correct. There are only 8 ways to write "SUPERMAN" in a circular fashion with "SUPER" as the substring, irrespective of the remaining letters. In each way, we have 3! additional ways to rearrange the remaining letters. Thus, 8 * 3!, which is 48.

Can someone help me see where my logic is incorrect?

• In these questions it is often the case that different rotations of a specific configuration are all considered the same. e.g. "SUPERMAN" and "UPERMANS" are the same arrangement. May 2, 2023 at 2:17
• It might be the problem is not clearly stated or understood. Does "SUPER" have to appear as a contiguous substring? Does the direction in which these letters appear matter? (the distinction between a necklace and a bracelet problem) May 2, 2023 at 2:47
• This question neglects to state what arrangements are considered equivalent, and for this reason cannot be answered. May 2, 2023 at 5:27

There are two ways to arrange the letters of the word SUPER as a continuous substring on a circle, clockwise and counterclockwise (anticlockwise). Relative to the word SUPER, there are $$3!$$ ways to arrange the letters of the word MAN as we proceed clockwise around the circle relative to the word SUPER. Hence, there are $$2 \cdot 3!$$ possible arrangements of the word SUPERMAN in which the letters of the word SUPER form a continuous substring.