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 of finding the probability that a word formed from the letters of "TITANIC" starts either with a T or a vowel, we can follow these steps:

### Step 1: Identify the Total Number of Arrangements
The word "TITANIC" consists of 7 letters where:
- T appears 2 times
- I appears 2 times
- A, N, C appear 1 time each

The total number of distinct arrangements of the letters can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \ldots} \]

Where \( n \) is the total number of letters, and \( n_1, n_2, n_3, \ldots \) are the frequencies of the distinct letters.

\[ \text{Total arrangements} = \frac{7!}{2! \times 2! \times 1! \times 1! \times 1!} = \frac{5040}{4} = 1260 \]

### Step 2: Count Favorable Outcomes
Next, we need to count the number of arrangements that start with either T or a vowel (which are A and I).

#### Case 1: Words starting with T
If the word starts with T, we have the remaining letters: T, I, I, A, N, C (6 letters).

The number of arrangements for these letters is:

\[ \text{Arrangements starting with T} = \frac{6!}{2! \times 1! \times 1! \times 1!} = \frac{720}{2} = 360 \]

#### Case 2: Words starting with A
If the word starts with A, we have the remaining letters: T, T, I, I, N, C (6 letters).

The number of arrangements for these letters is:

\[ \text{Arrangements starting with A} = \frac{6!}{2! \times 2! \times 1! \times 1!} = \frac{720}{4} = 180 \]

#### Case 3: Words starting with I
If the word starts with I, we have the remaining letters: T, T, I, A, N, C (6 letters).

The number of arrangements for these letters is:

\[ \text{Arrangements starting with I} = \frac{6!}{2! \times 1! \times 1! \times 1!} = \frac{720}{2} = 360 \]

### Step 3: Total Favorable Outcomes
Now, we sum the favorable outcomes from all cases:

\[ \text{Total favorable outcomes} = \text{Arrangements starting with T} + \text{Arrangements starting with A} + \text{Arrangements starting with I} \]
\[ = 360 + 180 + 360 = 900 \]

### Step 4: Calculate the Probability
The probability that a word formed starts either with T or a vowel is given by:

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total arrangements}} = \frac{900}{1260} \]

To simplify:

\[ \text{Probability} = \frac{15}{21} = \frac{5}{7} \]

### Final Answer
The probability that a word formed from the letters of "TITANIC" starts either with T or a vowel is \( \frac{5}{7} \).

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