Possible answers
A. X-Linked
NoIt is possible. However, it can'tis unlikely. The condition is claimed to be rare and as the sondaughter carries it, it means that the mother must have been a carrier. As the paternal grandmother also express the rarecondition, it means both of her parents must have been carrying it. It is therefore a little unlikely. We would mainly see the condition in men but much fewer cases in women.
B. Autosomal recessive
It is indeed possible. However asAs A
, it is unlikely though. As the condition is rare, it is unlikely that both the parents of the paternal grandmother were both carrying the mutant allele and also the mother of the two kids is carrying the mutant allele. I would tend to think it is therefore unlikely. If we had more information (such as the frequency of the disease and/or how many siblings bother the paternal grandmother has), then we could have a better opinion about the likelihood of "B." being the good answer.
C. Autosomal dominant
No, it can't be as none of the parents don't express the condition
D. Multifactorial
The term "multifactorial" is a little unclear to me. It might mean that there is environmental variance and/or it might mean that there are several loci that affect the inheritance of this condition (with eventually some epistasis). All of these are definitely possible and likely (see below).
E. Mitochondria
It is unlikely as 1) the disease is rare and 2) the father's mother expressed the disease. The father did not transmit the mtDNA (as mtDNA is inherited from the mother only).
What I would answer
For the vast majority of phenotypic conditions we consider, the inheritance is controlled by several SNPs and there is some non-zero environmental variance (that is, heritability is different from 1; see this post).
In my opinion, the question is hard to answer, not because it is a hard question but because it lacks information (and clarity). If I really had to answer, I would hence consider thanprobably answer "multifactorial" is most likely a good descriptor for this condition.