How many words can be formed with the letters of the word 'GANESHPURI' in which vowels occupy odd positions?

Answer

To solve the problem of how many words can be formed with the letters of the word "GANESHPURI" in which vowels occupy odd positions, we can follow these steps:

### Step 1: Identify the letters and their types
The word "GANESHPURI" consists of 10 letters:
- Vowels: A, E, U, I (4 vowels)
- Consonants: G, N, S, H, P, R (6 consonants)

### Step 2: Determine the positions available
In a 10-letter word, the positions are numbered from 1 to 10. The odd positions are 1, 3, 5, 7, and 9. Therefore, there are 5 odd positions available.

### Step 3: Choose positions for the vowels
We need to place the 4 vowels in the 5 odd positions. We can choose 4 positions out of the 5 available odd positions. The number of ways to choose 4 positions from 5 is given by the combination formula:

\[ \text{Number of ways to choose positions} = \binom{5}{4} = 5 \]

### Step 4: Arrange the vowels
Once we have chosen the positions for the vowels, we can arrange the 4 vowels (A, E, U, I) in those positions. The number of ways to arrange 4 vowels is given by:

\[ \text{Number of arrangements of vowels} = 4! = 24 \]

### Step 5: Place the consonants
After placing the vowels, we have 1 remaining odd position and 5 even positions (2, 4, 6, 8, 10) to fill with consonants. This gives us a total of 6 positions to fill with 6 consonants (G, N, S, H, P, R). The number of ways to arrange 6 consonants in these 6 positions is:

\[ \text{Number of arrangements of consonants} = 6! = 720 \]

### Step 6: Calculate the total arrangements
Now, we can find the total number of arrangements by multiplying the number of ways to choose the vowel positions, the arrangements of vowels, and the arrangements of consonants:

\[ \text{Total arrangements} = \binom{5}{4} \times 4! \times 6! = 5 \times 24 \times 720 \]

Calculating this gives:

\[ 5 \times 24 = 120 \]
\[ 120 \times 720 = 86400 \]

### Final Answer
Thus, the total number of words that can be formed with the letters of "GANESHPURI" in which vowels occupy odd positions is **86400**.

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