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
The word 'GANESHPURI' consists of the following letters:
- G, A, N, E, S, H, P, U, R, I

### Step 2: Count the Letters
The total number of letters in 'GANESHPURI' is 10.

### Step 3: Identify Vowels and Consonants
The vowels in the word are:
- A, E, U, I (4 vowels)

The consonants in the word are:
- G, N, S, H, P, R (6 consonants)

### Step 4: Identify Odd Positions
In a 10-letter word, the odd positions are:
- 1, 3, 5, 7, 9 (5 positions)

### Step 5: Place Vowels in Odd Positions
We have 4 vowels (A, E, U, I) and we need to place them in the 5 odd positions. Since we have one extra position, we can leave one of the odd positions blank.

### Step 6: Choose a Position to Leave Blank
We can choose any one of the 5 odd positions to leave blank. This can be done in \( \binom{5}{1} = 5 \) ways.

### Step 7: Arrange the Vowels
After choosing the blank position, we need to arrange the 4 vowels in the remaining 4 positions. The number of ways to arrange 4 vowels is \( 4! = 24 \).

### Step 8: Arrange the Consonants
Now, we have 5 even positions (2, 4, 6, 8, 10) to fill with the 6 consonants (G, N, S, H, P, R). We can choose any 5 out of these 6 consonants to fill the even positions. The number of ways to choose 5 consonants from 6 is \( \binom{6}{5} = 6 \).

### Step 9: Arrange the Chosen Consonants
The number of ways to arrange the 5 chosen consonants in the 5 even positions is \( 5! = 120 \).

### Step 10: Calculate the Total Arrangements
The total number of arrangements is given by multiplying the number of ways to choose the blank position, the arrangements of vowels, the ways to choose consonants, and the arrangements of consonants:
\[ \text{Total Arrangements} = \binom{5}{1} \times 4! \times \binom{6}{5} \times 5! \]
\[ = 5 \times 24 \times 6 \times 120 \]

### Step 11: Perform the Calculation
Calculating this gives:
\[ = 5 \times 24 = 120 \]
\[ 120 \times 6 = 720 \]
\[ 720 \times 120 = 86400 \]

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

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