How many words can be made from the word TING TING TRING in which vowels occupy even positions?

Options

A. 336000
B. 85360
C. 113600
D. none of these

Answer

To find how many words can be made from the word "TING TING TRING" in which vowels occupy even positions, we can follow these steps:

### Step 1: Identify the letters and their frequencies
The word "TING TING TRING" consists of the following letters:
- T: 3 times
- I: 3 times (the only vowel)
- N: 3 times
- G: 3 times
- R: 1 time

### Step 2: Determine the total number of letters
The total number of letters in "TING TING TRING" is 13.

### Step 3: Identify the even positions
In a 13-letter arrangement, the even positions are 2, 4, 6, 8, 10, and 12. This gives us a total of 6 even positions.

### Step 4: Place the vowels in the even positions
Since we have 3 vowels (I) and 6 even positions, we need to choose 3 out of these 6 positions to place the vowels. The number of ways to choose 3 positions from 6 is given by the combination formula:

\[ \text{Number of ways to choose positions} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = 20 \]

### Step 5: Arrange the vowels
Since all the vowels are identical (all are I), there is only 1 way to arrange them in the chosen positions.

### Step 6: Arrange the remaining letters
After placing the vowels, we have 10 positions left to fill with the remaining letters: T, T, T, N, N, N, G, G, G, R. The total number of arrangements of these letters is calculated using the formula for permutations of multiset:

\[ \text{Number of arrangements} = \frac{10!}{3! \times 3! \times 3! \times 1!} \]

Calculating this gives:

\[ 10! = 3628800 \]
\[ 3! = 6 \quad \text{(for T)} \]
\[ 3! = 6 \quad \text{(for N)} \]
\[ 3! = 6 \quad \text{(for G)} \]
\[ 1! = 1 \quad \text{(for R)} \]

So,

\[ \text{Number of arrangements} = \frac{3628800}{6 \times 6 \times 6 \times 1} = \frac{3628800}{216} = 16800 \]

### Step 7: Calculate the total arrangements
Now, we multiply the number of ways to choose the positions for the vowels by the number of arrangements of the remaining letters:

\[ \text{Total arrangements} = \binom{6}{3} \times \frac{10!}{3! \times 3! \times 3! \times 1!} = 20 \times 16800 = 336000 \]

### Final Answer
Thus, the total number of words that can be formed from "TING TING TRING" with vowels occupying even positions is **336000**.

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