How many words can be formed with the letters of the word PATALIPUTRA without changing the relative positions of vowels and consonants?

Options

A. a. 2170
B. b. 7200
C. c. 3600
D. d. 1040

Answer

To solve the problem of how many words can be formed with the letters of the word "PATALIPUTRA" without changing the relative positions of vowels and consonants, we can follow these steps:

### Step 1: Identify the Vowels and Consonants
The word "PATALIPUTRA" consists of the following letters:
- Vowels: A, A, A, I, U (5 vowels)
- Consonants: P, T, L, P, T, R (6 consonants)

### Step 2: Determine the Positions of Vowels and Consonants
In the word "PATALIPUTRA":
- The vowels are located at positions: 2, 4, 6, 8, 11
- The consonants are located at positions: 1, 3, 5, 7, 9, 10

### Step 3: Calculate the Arrangements of Vowels
The vowels A, A, A, I, U can be arranged. Since the letter 'A' repeats 3 times, the number of distinct arrangements of the vowels is calculated using the formula for permutations of multiset:

\[ \text{Arrangements of vowels} = \frac{5!}{3!} = \frac{120}{6} = 20 \]

### Step 4: Calculate the Arrangements of Consonants
The consonants P, T, L, P, T, R can also be arranged. Here, the letters 'P' and 'T' each repeat 2 times. Thus, the number of distinct arrangements of the consonants is:

\[ \text{Arrangements of consonants} = \frac{6!}{2! \times 2!} = \frac{720}{4} = 180 \]

### Step 5: Calculate the Total Arrangements
Since the arrangements of vowels and consonants are independent, we multiply the number of arrangements of vowels by the number of arrangements of consonants:

\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 20 \times 180 = 3600 \]

### Final Answer
The total number of words that can be formed with the letters of the word "PATALIPUTRA" without changing the relative positions of vowels and consonants is **3600**.

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