Final answer:
The specific gene rearrangement associated with follicular adenomas and follicular thyroid carcinomas typically involves a translocation between the PAX8 and PPARG genes, leading to the PAX8/PPARG fusion oncogene, which was not provided in the options.
Step-by-step explanation:
The gene rearrangement associated with follicular adenomas and follicular thyroid carcinomas is not the RET/PTC rearrangement, BCR-ABL fusion, or HER2 amplification. The most common genetic alteration in these conditions is a translocation involving the PAX8 gene and the PPARG gene. This chromosomal translocation results in the PAX8/PPARG fusion oncogene. However, this specific option is not listed among the provided choices. It is essential to note that KRAS mutations can be found in thyroid cancers, but they are more commonly associated with papillary thyroid carcinoma rather than with follicular adenomas or follicular carcinomas.