Let me clarify ...
So your A B C D E can take values 0 or 1. And a person can select "1" for more than one letter?
You could try row-binding A B C D E into "Y" and have a subject-wise random effect in addition to a letter-wise random effect.
I am assuming your data is structured as
| Person | A | B | C | D | E |
|---|---|---|---|---|---|
| 1001 | 0 | 0 | 1 | 0 | 1 |
| 1002 | 1 | 1 | 1 | 1 | 0 |
Convert it to something like
| Person | Letter | Y |
|---|---|---|
| 1001 | A | 0 |
| 1001 | B | 0 |
| 1001 | C | 1 |
| 1001 | D | 0 |
| 1001 | E | 1 |
| 1002 | A | 1 |
| 1002 | B | 1 |
| ... | ... | ... |
Then use glmer() as such
glmer(Y ~ <insert other X's> + (1 | Person) + (1 | Letter), family = "binomial", data = mydata)
Disclaimer though. There are many approaches to this problem but without knowing the context of what A B C D E mean, it's hard to give you a one method solve all approach. Even in the approach described above, there are different ways to specify the mixed effect model (e.g., do you really need the Letter-wise random effect? do you need more than random intercept? maybe random slope?). All of these things will have to be decided by the researcher with prior knowledge.