A DNA molecule in which both strands have radioactive thymidine is permitted to replicate in an environment that contains non-radioactive thymidine. What is the right number of DNA molecules which possess some radioactive thymidine post three duplications?

The correct option is "two such molecule".

To understand why this is correct, we can analyze the replication of DNA step-by-step using the semi-conservative model of DNA replication.
In semi-conservative replication, each of the two strands of a double-stranded DNA molecule serves as a template for the synthesis of a new complementary strand. Consequently, each resulting DNA double helix contains one original (parental) strand and one newly synthesized strand.

Let us represent the radioactive thymidine-containing strands (labeled strands) as R, and the non-radioactive thymidine-containing strands (unlabeled strands synthesized in the new environment) as N.

Initial State:
We start with one double-stranded DNA molecule where both strands are radioactive:
Parent DNA = R-R (contains 2 radioactive strands).

First Duplication (1st Replication Cycle):
The two radioactive strands (R and R) separate. Each template strand directs the synthesis of a new non-radioactive strand (N).
This yields:
- 1st molecule: R-N (possesses radioactive thymidine)
- 2nd molecule: R-N (possesses radioactive thymidine)
Total DNA molecules = 2, and both (2 molecules) possess some radioactive thymidine.

Second Duplication (2nd Replication Cycle):
The two R-N molecules replicate in the non-radioactive environment. The template strands separate: two R strands and two N strands. Each template strand gets paired with a newly synthesized non-radioactive strand (N).
This yields:
- From the first R-N: one R-N and one N-N
- From the second R-N: one R-N and one N-N
Total DNA molecules = 4. Out of these, 2 molecules (the R-N ones) possess some radioactive thymidine, while 2 molecules (the N-N ones) are completely non-radioactive.

Third Duplication (3rd Replication Cycle):
The four DNA molecules (two R-N and two N-N) replicate once more in the non-radioactive environment.
The template strands separating from these molecules are:
- Two R strands
- Six N strands (two from the R-N molecules and four from the two N-N molecules)
Each of these 8 template strands is paired with a newly synthesized non-radioactive strand (N).
This yields:
- 2 molecules containing one original radioactive strand: R-N
- 6 molecules containing only non-radioactive strands: N-N

Therefore, after three duplications, there are a total of 8 DNA molecules, out of which exactly two molecules possess some radioactive thymidine.