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'PROBLEMS' is an eight-letter word where none of the letters repeat.

We know that “$r$” objects, where all are distinct, can be reordered in $r!$ ways.

Therefore, if there had been no constraints, the letters of 'PROBLEMS' can be reordered in $8!$ ways.

However, the question states that '$P$' and '$S$' should occupy the first and last position respectively.

Therefore, the first and last position can be filled in only one way.

The remaining $6$ positions in between $P$ and $S$ can be filled with the $6$ letters in $6!$ ways.

Rearrangement Concepts: Number of ways to reorder objects:

- $r$ distinct objects can be reordered in $r!$ Ways.
- $r$ similar objects can be reordered in only $1$ way.
- 3 $r$ objects, of which $x$ are alike, can be reordered in $r!/x!$ ways.