A five letter word is to be formed taking all letters - H , A , R , E and T .What is the probability that the word formed will contain all the vowels together ?

Options

A. `(2)/(5)`
B. `(3)/(5)`
C. `(1)/(2)`
D. `(2)/(3)`

Answer

To solve the problem of finding the probability that a five-letter word formed from the letters H, A, R, E, and T contains all the vowels together, we can follow these steps:

### Step 1: Identify the letters and vowels
We have the letters: H, A, R, E, T. Among these, the vowels are A and E.

### Step 2: Calculate the total number of arrangements of the letters
To find the total number of arrangements of the five letters, we use the factorial of the number of letters:
\[ \text{Total arrangements} = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 3: Treat the vowels as a single unit
Since we want the vowels A and E to be together, we can treat them as a single entity or "block." Thus, we can represent the arrangement as:
- Block (AE or EA)
- H
- R
- T

This gives us four units to arrange: (AE), H, R, T.

### Step 4: Calculate the arrangements of the blocks
The number of arrangements of these four units is:
\[ \text{Arrangements of blocks} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 5: Arrange the vowels within their block
The vowels A and E can be arranged within their block in 2 ways (AE or EA):
\[ \text{Arrangements of vowels} = 2! = 2 \]

### Step 6: Calculate the total arrangements with vowels together
Now, we multiply the arrangements of the blocks by the arrangements of the vowels:
\[ \text{Total arrangements with vowels together} = 4! \times 2! = 24 \times 2 = 48 \]

### Step 7: Calculate the probability
The probability that the word formed will contain all the vowels together is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{48}{120} \]

### Step 8: Simplify the probability
Now, we simplify the fraction:
\[ \frac{48}{120} = \frac{12}{30} = \frac{2}{5} \]

Thus, the probability that the word formed will contain all the vowels together is:
\[ \text{Probability} = \frac{2}{5} \]

### Final Answer
The probability that the word formed will contain all the vowels together is \(\frac{2}{5}\).

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