Creating new columns in pandas dataframe as summed permutations of other columns

Use combinations instead permutations and for each values of tuple sum values together:

from itertools import combinations

cc = list(combinations(df.columns,3))
df = pd.concat([df.loc[:, c].sum(axis=1) for c in cc], axis=1, keys=cc)
df.columns = df.columns.map(''.join)
print (df)
   ABC  ABD  ABE  ACD  ACE  ADE  BCD  BCE  BDE  CDE
0    7    9    7   11    9   11   12   10   12   14
1    8    9   10    6    7    8    7    8    9    6
2   15   12   15    9   12    9   12   15   12    9

Using the underlying numpy array for efficiency.

from itertools import combinations

a = df.to_numpy()
df2 = pd.DataFrame(np.vstack([a[:,list(c)].sum(1)
                              for c in combinations(range(df.shape[1]), 3)
                              ]).T,
                   columns=map(''.join, combinations(df, 3))
                   )

output:

   ABC  ABD  ABE  ACD  ACE  ADE  BCD  BCE  BDE  CDE
0    7    9    7   11    9   11   12   10   12   14
1    8    9   10    6    7    8    7    8    9    6
2   15   12   15    9   12    9   12   15   12    9

NB. This is running ~10 times faster than the other pandas-based solution.

Or as an alternative:

import itertools
    
pd.DataFrame({'{}{}{}'.format(a, b, c): df[a] + df[b] + df[c] 
              for a, b, c in itertools.combinations(df.columns, 3)})

Output:

   ABC  ABD  ABE  ACD  ACE  ADE  BCD  BCE  BDE  CDE
0    7    9    7   11    9   11   12   10   12   14
1    8    9   10    6    7    8    7    8    9    6
2   15   12   15    9   12    9   12   15   12    9