Final answer:
The number of ways the letters of the word 'Constant' can be arranged without changing the relative positions of vowels and consonants is 540 because there are 3 ways to arrange the vowels and 180 ways to arrange the consonants. This makes answer (D) None of these correct because 540 is not listed as an option.
Step-by-step explanation:
The question seeks to find the number of ways the letters of the word 'Constant' can be arranged without changing the relative positions of the vowels and consonants. In the word 'Constant', the vowels present are 'o', 'a', and another 'a', and the consonants present are 'C', 'n', 's', 't', 'n', and 't'. The vowels need to stay in their positions relative to each other, as do the consonants.
We have a total of 3 vowels (oaa) and 4 distinct consonants (Cnst), with the letter 'n' and 't' each repeating once. The vowels can be arranged in 3! ways, but since there are two 'a's that are indistinguishable, we divide by 2!, which gives us 3!/2! = 3 arrangements.
The consonants can be arranged in 6!/2!2! = 720/4 = 180 ways due to repeats of 'n' and 't'. To get the total number of arrangements without changing the relative positions of vowels and consonants, we simply multiply the number of vowel arrangements by the number of consonant arrangements, which gives us 3 * 180 = 540.
Hence, the answer is (D) None of these, as 540 is not available in the option list provided.