A dictionary is made of the words that can be the position of the word "PARKAR" What is position of the word PARKAR in that dictionary if words are printed in the same order as that of ordinary dictionary?

Options

A. 98
B. 99
C. 100
D. 101

Answer

To find the position of the word "PARKAR" in a dictionary formed by all the permutations of its letters, we will follow these steps:

### Step 1: Arrange the letters in alphabetical order
The letters in "PARKAR" are A, A, K, P, R, R. When arranged in alphabetical order, they become:
- A, A, K, P, R, R

### Step 2: Count the total permutations starting with letters before 'P'
We need to calculate how many words can be formed starting with letters that come before 'P' in the alphabetical order.

#### Words starting with 'A':
- If 'A' is the first letter, the remaining letters are A, K, P, R, R.
- The number of permutations is given by:

\[ \frac{5!}{2! \cdot 2!} = \frac{120}{4} = 30 \]

#### Words starting with 'K':
- If 'K' is the first letter, the remaining letters are A, A, P, R, R.
- The number of permutations is given by:

\[ \frac{5!}{2! \cdot 2!} = \frac{120}{4} = 30 \]

So, the total number of words starting with letters before 'P' is:

\[ 30 (from A) + 30 (from K) = 60 \]

### Step 3: Count the permutations starting with 'P'
Now we consider words starting with 'P'. The next letter in "PARKAR" is 'A'.

#### Words starting with 'PA':
- If 'PA' is the first two letters, the remaining letters are A, K, R, R.
- The number of permutations is given by:

\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

#### Words starting with 'PK':
- If 'PK' is the first two letters, the remaining letters are A, A, R, R.
- The number of permutations is given by:

\[ \frac{4!}{2! \cdot 2!} = \frac{24}{4} = 6 \]

#### Words starting with 'PR':
- If 'PR' is the first two letters, the remaining letters are A, A, K, R.
- The number of permutations is given by:

\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 4: Count the permutations starting with 'PAR'
Now we consider words starting with 'PAR'. The next letter in "PARKAR" is 'K'.

#### Words starting with 'PARA':
- If 'PARA' is the first four letters, the remaining letters are K, R.
- The number of permutations is given by:

\[ 2! = 2 \]

#### Words starting with 'PARK':
- If 'PARK' is the first four letters, the remaining letters are A, R.
- The number of permutations is given by:

\[ 2! = 2 \]

### Step 5: Count the position of "PARKAR"
Now we can sum up all the permutations counted:

1. Words starting with A: 30
2. Words starting with K: 30
3. Words starting with PA: 12
4. Words starting with PK: 6
5. Words starting with PR: 12
6. Words starting with PARA: 2
7. Words starting with PARK: 2
8. Finally, "PARKAR" itself counts as the next position.

Adding these up:

\[ 30 + 30 + 12 + 6 + 12 + 2 + 2 + 1 = 95 \]

Thus, the position of the word "PARKAR" in the dictionary is **96**.

### Final Answer:
The position of the word "PARKAR" in the dictionary is **96**.

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