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## Main Question or Discussion Point

Hello, I am a philosophy student. I have no extensive training at math so please don't deluge me with formalism or terms of art off the bat. I also know that I run the risk of asking a really basic question here, but forgive me if so.

In my studies, I am often confronted with the need to enumerate the possibilities given the following circumstances.

Take, say, the ordered string of symbols XYZ.

Now take the square bracket, [..] which can go around any of the symbols or any ordered string of the symbols.

The rules are the square bracket must go around at least one of the symbols.

I get the following possibilities:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

X[YZ]

[XYZ]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[X][YZ}

[X][Y][Z]

Can someone please explain the general pattern or formula that allows one to calculate the number of these possibilities from an n-ly long string of symbols? So a triply long string of symbols nets you 12 possibilities.

Update***

I would also like to know about the pattern that would get one the following set of possibilities:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

[[X]Y]Z

[X[Y]]Z

X[YZ]

X[[Y]Z]

X[Y[Z]]

[XYZ]

[[X]YZ]

[X[Y]Z]

[XY[Z]]

[[XY]Z]

[[[X]Y]Z]

[[X[Y]]Z]

[X[YZ]]

[X[[Y]Z]]

[X[Y[Z]]]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[[X]Y][Z]

[X[Y]][Z]

[X][YZ]

[X][[Y]Z]

[X][Y[Z]]

[X][Y][Z]

and:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

[[X]Y]Z

[X[Y]]Z

X[YZ]

X[[Y]Z]

X[Y[Z]]

[XYZ]

[[X]YZ]

[X[Y]Z]

[XY[Z]]

[[XY]Z]

[[XY][Z]]

[[[X]Y]Z]

[[[X]Y][Z]]

[[X[Y]]Z]

[[X[Y]][Z]]

[X[YZ]]

[[X][YZ]]

[X[[Y]Z]]

[[X][[Y]Z]]

[X[Y[Z]]]

[[X][Y[Z]]]

[[X][Y]Z]

[[X]Y[Z]]

[X[Y][Z]]

[[X][Y][Z]]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[[X]Y][Z]

[X[Y]][Z]

[X][YZ]

[X][[Y]Z]

[X][Y[Z]]

[X][Y][Z]

I might have missed some of these but hopefully the pattern is reasonably clear.

In my studies, I am often confronted with the need to enumerate the possibilities given the following circumstances.

Take, say, the ordered string of symbols XYZ.

Now take the square bracket, [..] which can go around any of the symbols or any ordered string of the symbols.

The rules are the square bracket must go around at least one of the symbols.

I get the following possibilities:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

X[YZ]

[XYZ]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[X][YZ}

[X][Y][Z]

Can someone please explain the general pattern or formula that allows one to calculate the number of these possibilities from an n-ly long string of symbols? So a triply long string of symbols nets you 12 possibilities.

Update***

I would also like to know about the pattern that would get one the following set of possibilities:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

[[X]Y]Z

[X[Y]]Z

X[YZ]

X[[Y]Z]

X[Y[Z]]

[XYZ]

[[X]YZ]

[X[Y]Z]

[XY[Z]]

[[XY]Z]

[[[X]Y]Z]

[[X[Y]]Z]

[X[YZ]]

[X[[Y]Z]]

[X[Y[Z]]]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[[X]Y][Z]

[X[Y]][Z]

[X][YZ]

[X][[Y]Z]

[X][Y[Z]]

[X][Y][Z]

and:

[X]YZ

X[Y]Z

XY[Z]

[XY]Z

[[X]Y]Z

[X[Y]]Z

X[YZ]

X[[Y]Z]

X[Y[Z]]

[XYZ]

[[X]YZ]

[X[Y]Z]

[XY[Z]]

[[XY]Z]

[[XY][Z]]

[[[X]Y]Z]

[[[X]Y][Z]]

[[X[Y]]Z]

[[X[Y]][Z]]

[X[YZ]]

[[X][YZ]]

[X[[Y]Z]]

[[X][[Y]Z]]

[X[Y[Z]]]

[[X][Y[Z]]]

[[X][Y]Z]

[[X]Y[Z]]

[X[Y][Z]]

[[X][Y][Z]]

[X][Y]Z

X[Y][Z]

[X]Y[Z]

[XY][Z]

[[X]Y][Z]

[X[Y]][Z]

[X][YZ]

[X][[Y]Z]

[X][Y[Z]]

[X][Y][Z]

I might have missed some of these but hopefully the pattern is reasonably clear.

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