Number of words with alternate vowels and consonants

To solve this problem, we need to first identify the vowels and consonants in the given word.

APURNA
Vowels: A, U
Consonants: P, R, N

Now, we need to arrange these vowels and consonants in such a way that they appear alternate. We can start with placing a vowel (A or U) at the first position, and then alternate between vowels and consonants. We can do this by following these steps:

Step 1: Choose a vowel for the first position (2 choices)
Step 2: Choose a consonant for the second position (3 choices)
Step 3: Choose a vowel for the third position (1 choice, since we used one vowel already)
Step 4: Choose a consonant for the fourth position (2 choices, since we used one consonant already)
Step 5: Choose a vowel for the fifth position (1 choice)
Step 6: Choose a consonant for the sixth position (1 choice)

We can then multiply the number of choices for each step to get the total number of words:

Total number of words = 2 x 3 x 1 x 2 x 1 x 1 = 12

However, we can also choose the other vowel (U or A) for the first position, and this will give us another set of 12 words. Therefore, the total number of words is:

Total number of words = 2 x 12 = 24

However, we have overcounted some words, since we can arrange the three consonants in 3! = 6 ways, and the two vowels in 2! = 2 ways. Therefore, we need to divide the total number of words by (3! x 2!) to get the final answer:

Final answer = 24 / (3! x 2!) = 24 / 12 = 2 x 2 x 1 = 4

Therefore, the number of words that can be made by rearranging the letters of the word APURNA so that vowels and consonants appear alternate is 4. The correct option is C.