What is the probability of formating word from the letters of word"TUESDAY" such that all vowels come together ?

Options

A. `(8)/(35)`
B. `(1)/(7)`
C. `(3)/(35)`
D. `(17)/(35)`

Explanation

To find the probability of forming words from the letters of the word "TUESDAY" such that all vowels come together, we will follow these steps:

### Step 1: Identify the letters in "TUESDAY"
The word "TUESDAY" consists of the following letters:
- Consonants: T, S, D, Y
- Vowels: U, E, A

### Step 2: Treat all vowels as a single unit
Since we want all vowels to come together, we can treat the group of vowels (U, E, A) as a single unit or block. Therefore, we can represent the letters as:
- Block of vowels: (UEA)
- Consonants: T, S, D, Y

Now we have the following units to arrange:
- (UEA), T, S, D, Y

### Step 3: Count the total units
We have a total of 5 units to arrange:
1. (UEA)
2. T
3. S
4. D
5. Y

### Step 4: Calculate the arrangements of these units
The number of ways to arrange these 5 units is given by the factorial of the number of units:
\[ 5! = 120 \]

### Step 5: Arrange the vowels within their block
The vowels U, E, A can be arranged among themselves. The number of arrangements of the 3 vowels is:
\[ 3! = 6 \]

### Step 6: Calculate the total arrangements with vowels together
Now, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements with vowels together} = 5! \times 3! = 120 \times 6 = 720 \]

### Step 7: Calculate the total arrangements of the letters in "TUESDAY"
The total number of arrangements of the letters in "TUESDAY" (which has 7 distinct letters) is:
\[ 7! = 5040 \]

### Step 8: Calculate the probability
The probability that all vowels come together is given by the ratio of the number of favorable arrangements to the total arrangements:
\[ P(\text{vowels together}) = \frac{\text{Total arrangements with vowels together}}{\text{Total arrangements of letters}} = \frac{720}{5040} \]

### Step 9: Simplify the probability
To simplify:
\[ P(\text{vowels together}) = \frac{720 \div 720}{5040 \div 720} = \frac{1}{7} \]

### Final Answer
The probability of forming words from the letters of the word "TUESDAY" such that all vowels come together is \( \frac{1}{7} \).

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