- #1
mcintyw2
- 2
- 0
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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