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.

Wrong answer because this question highlighted rearrange the ways so that 1should be subtracted from 36 answer should be 35