If the letters of word ASSASSIN are written at random in a raw, then which of following are true?

Options

A. Probability that all 'S' come together is `1/14`
B. Probability that all 'S' do not come together is `1/7`
C. Probability that no two 'S' come together is `1/14`
D. Probability that vowels occupy places of vowel only is `1/56`

Answer

To solve the problem, we need to analyze the word "ASSASSIN" and find the probabilities for the different conditions specified in the question. Let's break down the solution step by step.

### Step 1: Total Arrangements of the Word "ASSASSIN"
The word "ASSASSIN" consists of 8 letters: A, S, S, A, S, S, I, N.

- The letters consist of:
- A: 2 times
- S: 4 times
- I: 1 time
- N: 1 time

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

\[ \text{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

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

\[ \text{Total arrangements} = \frac{8!}{2! \cdot 4! \cdot 1! \cdot 1!} \]

Calculating this gives:

\[ 8! = 40320, \quad 2! = 2, \quad 4! = 24 \]

So,

\[ \text{Total arrangements} = \frac{40320}{2 \cdot 24 \cdot 1 \cdot 1} = \frac{40320}{48} = 840 \]

### Step 2: Probability that All S's Come Together
To find the probability that all S's come together, we can treat the four S's as a single block. Thus, we have the block "SSSS" and the letters A, A, I, N.

Now we have 5 items to arrange: SSSS, A, A, I, N.

The arrangements of these 5 items are:

\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

Now, the probability that all S's come together is:

\[ P(\text{All S's together}) = \frac{\text{Arrangements with S's together}}{\text{Total arrangements}} = \frac{60}{840} = \frac{1}{14} \]

### Step 3: Probability that All S's Do Not Come Together
The probability that all S's do not come together is simply:

\[ P(\text{All S's do not come together}) = 1 - P(\text{All S's together}) = 1 - \frac{1}{14} = \frac{13}{14} \]

### Step 4: Probability that No Two S's Come Together
To find the probability that no two S's come together, we can arrange the letters A, A, I, N first, which gives us:

\[ \text{Arrangements of A, A, I, N} = \frac{4!}{2!} = 12 \]

Now we have 4 letters (A, A, I, N) arranged, creating 5 gaps (before the first letter, between letters, and after the last letter) to place the S's:

- _ A _ A _ I _ N _

We need to choose 4 out of these 5 gaps to place the S's, which can be done in:

\[ \text{Ways to choose gaps} = \binom{5}{4} = 5 \]

Thus, the total arrangements where no two S's come together is:

\[ \text{Total arrangements} = 12 \times 5 = 60 \]

The probability that no two S's come together is:

\[ P(\text{No two S's together}) = \frac{60}{840} = \frac{1}{14} \]

### Step 5: Probability that Vowels Occupy Places of Vowels Only
The vowels in "ASSASSIN" are A, A, I. We need to arrange these vowels in their respective positions.

The arrangements of the vowels A, A, I are:

\[ \text{Arrangements of A, A, I} = \frac{3!}{2!} = 3 \]

Now, the remaining letters (S, S, S, S, N) can be arranged in:

\[ \text{Arrangements of S, S, S, S, N} = \frac{5!}{4!} = 5 \]

Thus, the total arrangements where vowels occupy places of vowels only is:

\[ \text{Total arrangements} = 3 \times 5 = 15 \]

The probability that vowels occupy places of vowels only is:

\[ P(\text{Vowels in vowel places}) = \frac{15}{840} = \frac{1}{56} \]

### Summary of Results
1. Probability that all S's come together: \( \frac{1}{14} \)
2. Probability that all S's do not come together: \( \frac{13}{14} \)
3. Probability that no two S's come together: \( \frac{1}{14} \)
4. Probability that vowels occupy places of vowels only: \( \frac{1}{56} \)