Introduction:
The problem requires us to find the number of words that can be made by rearranging the letters of the word APURNA such that vowels and consonants appear alternately.

Approach:
To solve this problem, we need to follow the following steps:

1. Separate the vowels and consonants in the given word.
2. Determine the number of ways the vowels and consonants can be arranged.
3. Multiply the number of ways the vowels and consonants can be arranged to get the total number of words.

Step 1:
Separate the vowels and consonants in the word APURNA:
- Vowels: A, U
- Consonants: P, R, N

Step 2:
Determine the number of ways the vowels and consonants can be arranged:
- There are 2 vowels and 3 consonants, so we have two cases:
- Case 1: Vowel-consonant-vowel-consonant-vowel
- Number of ways to arrange the vowels: 2! = 2
- Number of ways to arrange the consonants: 3! = 6
- Total number of ways: 2 x 6 = 12
- Case 2: Consonant-vowel-consonant-vowel-consonant
- Number of ways to arrange the consonants: 3! = 6
- Number of ways to arrange the vowels: 2! = 2
- Total number of ways: 6 x 2 = 12

Step 3:
Multiply the number of ways the vowels and consonants can be arranged to get the total number of words:
- Total number of words = 12 + 12 = 24

Conclusion:
In conclusion, the total number of words that can be made by rearranging the letters of the word APURNA such that vowels and consonants appear alternately is 24.

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