Question 880687
You used combinations rather than permutations;
so you are finding the number of sets (order does not matter) of 3 letters including at least one vowel, no repetitions allowed.
One of your sets is {A,B,C}.
However, the set {A,B,C} is shared by {{{6}}} different codes.
Codes ABC, ACB, BAC, BCA, CAB, and CBA are considered different codes, because they are different permutations of the elements of the set, and for a code order matters.
Each of your {{{1270}}} sets of 3 letters generates {{{3!=6}}} codes,
and {{{1270*6=7620}}} .
 
The result {{{7620}}} requires that it be specified that repetitions are not allowed. (If repetitions were allowed, as usually happens for codes< the result would be a higher number, but the problem would not be so interesting).
That result {{{7620}}} can be reached most directly by
subtracting the number of 3-consonant no-repeat codes, {{{21*20*19=7980}}} ,
from the number of 3-letter no-repeat codes, {{{26*25*24=15600}}} .