Solution:

To solve this problem, we need to find the total number of arrangements of the letters of the words CALCUTTA and AMERICA separately, and then find the ratio of the two numbers.

Arrangements of CALCUTTA:

There are 8 letters in the word CALCUTTA, out of which there are 2 C's, 2 T's, and 2 A's.

The total number of arrangements of the letters of CALCUTTA can be found using the formula for permutations with repetition, which is:

n! / (r1! r2! ... rk!)

where n is the total number of objects, and r1, r2, ..., rk are the frequencies of the k distinct objects.

In this case, n = 8, r1 = r2 = r3 = 2, and all other frequencies are 1. Therefore, the number of arrangements of CALCUTTA is:

8! / (2! 2! 2!) = 28 * 7 * 6 = 1176

Arrangements of AMERICA:

There are 7 letters in the word AMERICA, out of which there are 2 A's and 2 E's.

The total number of arrangements of the letters of AMERICA can be found using the formula for permutations with repetition, which is:

n! / (r1! r2! ... rk!)

In this case, n = 7, r1 = r2 = 2, and all other frequencies are 1. Therefore, the number of arrangements of AMERICA is:

7! / (2! 2!) = 1260

Ratio of the number of arrangements:

The ratio of the number of arrangements of CALCUTTA to the number of arrangements of AMERICA is:

1176 / 1260 = 14 / 15

Therefore, the answer is none of these, as the ratio is not 1:2 or 2:1 or 2:2.