The probability that in a random arrangement of the letter of the word "FAVOURABLE" the two 'A' do not come together is

Options

A. `1/5`
B. `1/10`
C. `9/10`
D. `4/5`

Answer

To find the probability that the two 'A's in the word "FAVOURABLE" do not come together, we can follow these steps:

### Step 1: Determine the total number of letters and arrangements
The word "FAVOURABLE" has 10 letters in total, with the letter 'A' repeating twice. The total number of arrangements of the letters can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{10!}{2!} \]

### Step 2: Calculate the total arrangements
Calculating this gives:

\[ 10! = 3628800 \quad \text{and} \quad 2! = 2 \]

Thus,

\[ \text{Total arrangements} = \frac{3628800}{2} = 1814400 \]

### Step 3: Calculate arrangements where the two 'A's come together
When the two 'A's are considered as a single entity (or letter), we can treat them as one letter. Therefore, we now have the following letters to arrange: {AA, F, V, O, U, R, B, L, E}, which gives us a total of 9 letters.

The number of arrangements of these 9 letters is:

\[ \text{Arrangements with 'A's together} = 9! \]

### Step 4: Calculate the arrangements with 'A's together
Calculating this gives:

\[ 9! = 362880 \]

### Step 5: Calculate the probability that the two 'A's come together
Now, the probability that the two 'A's come together is given by the ratio of the arrangements where 'A's are together to the total arrangements:

\[ P(\text{A's together}) = \frac{9!}{\frac{10!}{2!}} = \frac{362880}{1814400} \]

### Step 6: Simplify the probability
Calculating this gives:

\[ P(\text{A's together}) = \frac{1}{5} \]

### Step 7: Calculate the probability that the two 'A's do not come together
To find the probability that the two 'A's do not come together, we subtract the probability that they do come together from 1:

\[ P(\text{A's not together}) = 1 - P(\text{A's together}) = 1 - \frac{1}{5} = \frac{4}{5} \]

### Final Answer
Thus, the probability that the two 'A's do not come together is:

\[ \frac{4}{5} \]

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