Letters of the word TITANIC are arranged to form all possible words. What is the probability that a word formed starts either with a T or a vowel ?

Options

A. `(2)/(7)`
B. `(4)/(7)`
C. `(3)/(7)`
D. `(5)/(7)`

Answer

To solve the problem, we need to calculate the probability that a word formed from the letters of the word "TITANIC" starts with either a 'T' or a vowel (which are 'A' and 'I').

### Step-by-Step Solution:

1. **Identify the total number of letters and their repetitions**:
The word "TITANIC" consists of 7 letters: T, I, T, A, N, I, C.
Here, T appears 2 times, I appears 2 times, and A, N, C appear 1 time each.

2. **Calculate the total arrangements of the letters**:
The total number of arrangements of the letters in "TITANIC" can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{7!}{2! \times 2!} \]
where \(7!\) is the factorial of the total letters, and \(2!\) accounts for the repetitions of T and I.

3. **Calculate the total arrangements**:
\[ 7! = 5040 \]
\[ 2! = 2 \quad \text{(for T)} \]
\[ 2! = 2 \quad \text{(for I)} \]
Therefore,
\[ \text{Total arrangements} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260 \]

4. **Calculate the favorable outcomes**:
We need to find the arrangements that start with either 'T', 'A', or 'I'.

- **Case 1: Starts with T**:
If the first letter is T, the remaining letters are I, T, A, N, I, C (6 letters).
The arrangements are:
\[ \text{Arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

- **Case 2: Starts with I**:
If the first letter is I, the remaining letters are T, T, A, N, I, C (6 letters).
The arrangements are:
\[ \text{Arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

- **Case 3: Starts with A**:
If the first letter is A, the remaining letters are T, I, T, N, I, C (6 letters).
The arrangements are:
\[ \text{Arrangements} = \frac{6!}{2! \times 2!} = \frac{720}{4} = 180 \]

5. **Total favorable outcomes**:
Now, we sum the favorable outcomes from all cases:
\[ \text{Total favorable outcomes} = 360 + 360 + 180 = 900 \]

6. **Calculate the probability**:
The probability that a word starts with either a T or a vowel is given by:
\[ P(\text{Starts with T or a vowel}) = \frac{\text{Total favorable outcomes}}{\text{Total arrangements}} = \frac{900}{1260} \]

7. **Simplify the probability**:
\[ P = \frac{900 \div 180}{1260 \div 180} = \frac{5}{7} \]

### Final Answer:
The probability that a word formed starts with either a T or a vowel is \(\frac{5}{7}\).