Cooperative apartment houses have the peculiar distinction of being

The sentence is simple and all it is trying to convey is that cooperative apartment houses are dwellings (residential) that must operate as businesses too.
Avid readers would recognize that ‘distinction of’ (or ‘distinction between A and B’) is a lot more natural than ‘distinction that …,’ and we certainly do not write ‘distinction to be.’ So, options (A) and (B) are preferable over (C), (D) and (E). But we don’t need to rely on knowing this distinction to get to the correct answer.

(A) of being dwellings that must also operate as businesses
(B) of dwellings that must also operate like business

When we use ‘distinction of,’ we must follow the preposition ‘of’ with a noun. In option (A), ‘being’ is used as a gerund and is correct. When we remove ‘being’ as done in option (B), it seems to say that cooperative apartment houses have the distinction of ‘dwellings that must also operate as businesses’ i.e. whatever is the distinction of these dwellings, that is the distinction of cooperative apartment houses too. But that is not correct. What we want to convey is that they are dwellings and must also operate as businesses.

Hence, option (B) doesn’t make sense.

(C) that they are dwellings that must operate like business
(D) that, as dwellings, they must also operate like businesses

The dwellings must operate ‘as (in the role of) businesses,’ not ‘like businesses.’ We are not comparing the dwellings with businesses. And if we were to compare the operations of the two, we need to use ‘as’ with a verb ‘dwellings must operate as businesses do.’ But this is also not what we mean. We mean that dwellings must operate in the role of a business and as discussed in our module, ‘as’ plays that role too.

For example,
He works as a manager. (in the role of/ in the capacity of)

Hence, options (B), (C) and (D) are not correct.

Option (D) uses ‘as dwellings’ as a parenthetical expression which doesn’t make sense in our context. We need to say that the houses ‘are dwellings’ (not ‘as dwellings’) that must function as businesses too.

(E) to be a dwelling that must also operate as a business

Another problem is that we have used plural ‘houses’ in the non – underlined part. So we should use plural ‘dwellings’ and plural ‘businesses.’ Hence options (B), (C) and (E) are not correct.

Finally, we are left with option (A) which has no errors.

Answer (A)