The number of arrangement s of the letter of the word PAPAYA in which the two 'P' do not appear adjacently is

Options

A. 40
B. 60
C. 80
D. 36

Answer

To find the number of arrangements of the letters of the word "PAPAYA" in which the two 'P's do not appear adjacently, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "PAPAYA".

The word "PAPAYA" consists of 6 letters: P, A, P, A, Y, A. Here, the letter 'A' appears 3 times, and the letter 'P' appears 2 times.

The formula for the total arrangements of letters when there are repeating letters is given by:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

Where:
- \( n \) is the total number of letters.
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeating letters.

In our case:

- Total letters, \( n = 6 \)
- Frequency of 'A', \( n_1 = 3 \)
- Frequency of 'P', \( n_2 = 2 \)

Thus, the total arrangements can be calculated as:

\[ \text{Total arrangements} = \frac{6!}{3! \times 2!} \]

Calculating this gives:

\[ 6! = 720, \quad 3! = 6, \quad 2! = 2 \]

So,

\[ \text{Total arrangements} = \frac{720}{6 \times 2} = \frac{720}{12} = 60 \]

### Step 2: Calculate the arrangements where the two 'P's are together.

To find the arrangements where the two 'P's are together, we can treat the two 'P's as a single block. This gives us the following letters to arrange: (PP), A, A, A, Y.

Now we have 5 letters: (PP), A, A, A, Y. Here, 'A' appears 3 times.

Using the same formula for arrangements:

\[ \text{Arrangements with P together} = \frac{5!}{3!} \]

Calculating this gives:

\[ 5! = 120, \quad 3! = 6 \]

So,

\[ \text{Arrangements with P together} = \frac{120}{6} = 20 \]

### Step 3: Subtract the arrangements with 'P' together from the total arrangements.

Now, we can find the number of arrangements where the two 'P's do not appear together by subtracting the arrangements where they are together from the total arrangements:

\[ \text{Arrangements with P not together} = \text{Total arrangements} - \text{Arrangements with P together} \]

Substituting the values we calculated:

\[ \text{Arrangements with P not together} = 60 - 20 = 40 \]

### Final Answer

Thus, the number of arrangements of the letters of the word "PAPAYA" in which the two 'P's do not appear adjacently is **40**.

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