How many possible ways can this be done? (A,B,C,D,E)

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SUMMARY

The discussion centers on the combinatorial problem of arranging the letters A, B, C, D, and E in various configurations. It is established that there are exactly 31 unique ways to arrange these letters, calculated using the formula $2^5 - 1$, which accounts for all possible combinations excluding the empty set. The participants confirm that the problem is a classic example of combinatorial enumeration, specifically focusing on subsets of a set.

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Example:
1. (A,B,C,D,E)
2. (A,B,C,D)
3. (A,B,C)
4. (A,B)
5. (A)
6. (A,B,D,E)
etc...
I just put it in a table format, as it's easier to count this way (31 columns = 31 solutions).
Not sure if there are any more ways to do this.

Questions:

1) Are there only 31 ways to do this?

2) Is there like a calculator online which solves this sort of problem?

3) Is there a name for this sort of problem?
 
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It appears you want A, B, C, D and E arranged in alphabetical order with at least one letter present. There are $2^5-1 = 31$ ways to do this; we subtract $1$ to omit the configuration of having all blanks. The $2^5$ comes from the number of possibilities for each "slot", which is $2$: a letter or a blank.
 

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