Yes, it has a lower frequency
<span>1 archaic : happy, pleased
2 archaic : inclined, desirous
3 a : willing
he was very fain, for the young widow was “altogether fair and lovely … ” — Amy Kelly
b : being obliged or constrained : compelled
Great Britain was fain to devote its whole energy … to the business of slaying and being slain — G. M. Trevelyan</span>
I believe the answer is "C. to carry" (correct me if im wrong)