The correct answer is option 'A': III, II, I, IV, V, VI

Marco Polo, Ibn-Batuta, and Nikitin were three foreign travelers who visited India during different periods. Let's arrange them in chronological order along with the other travelers mentioned:

I. Nikitin
II. Ibn-Batuta
III. Marco Polo
IV. Nicolo Conti
V. Abdur Razzaq
VI. Barbosa

Now let's discuss each traveler and their respective time periods:

**Marco Polo (1254-1324)**:
Marco Polo was an Italian merchant and explorer who is famous for his travels to Asia, including India. He embarked on his journey in 1271 and reached India in 1292. He spent several years in India, exploring various regions and documenting his observations. Marco Polo's travels took place during the 13th century.

**Ibn-Batuta (1304-1368/1377)**:
Ibn-Batuta was a Moroccan scholar and explorer who embarked on a journey in 1325 to perform the Hajj pilgrimage. His journey extended over several decades, and he covered a vast territory, including India. Ibn-Batuta visited India multiple times between 1333 and 1347, spending a significant amount of time in the region. His travels occurred during the 14th century.

**Nikitin (1466-1474)**:
Nikitin was a Russian merchant who traveled to India during the 15th century. He arrived in India around 1468 and spent several years trading and exploring the country. Nikitin's travels took place during the 15th century.

**Nicolo Conti (1395-1469)**:
Nicolo Conti was an Italian merchant who visited India during the 15th century. He arrived in India around 1420 and spent several years exploring various regions, including the Malabar Coast. Conti's travels occurred during the 15th century.

**Abdur Razzaq (1413-1482)**:
Abdur Razzaq was an ambassador from the Timurid Empire who visited India in the 15th century. He traveled to India in the early 1440s and stayed for a few years. Razzaq's travels took place during the 15th century.

**Barbosa (1480-1521)**:
Barbosa was a Portuguese explorer who visited India during the early 16th century. He arrived in India in 1500 and spent several years documenting his observations and experiences in the country. Barbosa's travels occurred during the 16th century.

Therefore, the correct chronological order of the foreign travelers is:

III. Marco Polo
II. Ibn-Batuta
I. Nikitin
IV. Nicolo Conti
V. Abdur Razzaq
VI. Barbosa

Hence, option 'A' is the correct answer.