The letters of the word "SOCIETY" are placed at random in a row. The probability that three vowels occur together is

Options

A. `1/7`
B. `2/7`
C. `3/7`
D. `4/7`

Answer

To find the probability that the three vowels in the word "SOCIETY" occur together, we can follow these steps:

### Step 1: Identify the letters in the word "SOCIETY"
The word "SOCIETY" consists of 7 letters: S, O, C, I, E, T, Y.

### Step 2: Count the vowels and consonants
In the word "SOCIETY":
- Vowels: O, I, E (Total = 3 vowels)
- Consonants: S, C, T, Y (Total = 4 consonants)

### Step 3: Treat the three vowels as a single unit
To find the arrangements where the vowels are together, we can treat the three vowels (O, I, E) as one single unit or letter. Thus, we can represent the arrangement as:
- (OIE), S, C, T, Y

Now we have 5 units to arrange: (OIE), S, C, T, Y.

### Step 4: Calculate the arrangements of the units
The number of ways to arrange these 5 units is given by:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 5: Calculate the arrangements of the vowels within the unit
The three vowels (O, I, E) can be arranged among themselves in:
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 6: Calculate the total arrangements with vowels together
The total arrangements where the vowels are together is:
\[ 5! \times 3! = 120 \times 6 = 720 \]

### Step 7: Calculate the total arrangements of the letters in "SOCIETY"
The total number of arrangements of the 7 letters in "SOCIETY" is:
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

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

### Step 9: Simplify the probability
Now, simplifying the fraction:
\[ \frac{720}{5040} = \frac{1}{7} \]

### Final Answer
Thus, the probability that the three vowels occur together is:
\[ \frac{1}{7} \]

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