A dictionary is printed consisting of 7 lettered words only that can be made with letters of the word ''CRICKET''. If the words are printed in the alphabetical order, as in the ordinary dictionary, then the number of words before the word CRICKET, is

Options

A. 530
B. 480
C. 531
D. 481

Answer

To find the number of words that come before the word "CRICKET" in a dictionary made from its letters, we will follow these steps:

### Step 1: List the letters of "CRICKET" in alphabetical order
The letters of "CRICKET" are C, R, I, C, K, E, T. Arranging these in alphabetical order gives us:
- C, C, E, I, K, R, T

### Step 2: Count the total permutations of the letters
Since the word "CRICKET" has repeated letters (C appears twice), the total number of distinct permutations of the letters can be calculated using the formula for permutations of multiset:
\[ \text{Total permutations} = \frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!} \]
where \( n \) is the total number of letters, and \( n_1, n_2, ... n_k \) are the frequencies of the distinct letters.

Here, we have:
- Total letters (n) = 7 (C, C, E, I, K, R, T)
- Frequencies: C = 2, E = 1, I = 1, K = 1, R = 1, T = 1

Thus, the total permutations are:
\[ \frac{7!}{2!} = \frac{5040}{2} = 2520 \]

### Step 3: Count words starting with letters before 'C'
Since "CRICKET" starts with 'C', we will count the words that start with letters that come before 'C'. However, in this case, there are no letters before 'C' in the sorted order.

### Step 4: Count words starting with 'C'
Next, we need to count the words that start with 'C'. The remaining letters are C, E, I, K, R, T.

#### Step 4.1: Count words starting with 'CC'
If the word starts with 'CC', the remaining letters are E, I, K, R, T. The number of permutations of these 5 letters is:
\[ 5! = 120 \]

#### Step 4.2: Count words starting with 'CE'
If the word starts with 'CE', the remaining letters are C, I, K, R, T. The number of permutations of these letters is:
\[ 5! = 120 \]

#### Step 4.3: Count words starting with 'CI'
If the word starts with 'CI', the remaining letters are C, E, K, R, T. The number of permutations of these letters is:
\[ 5! = 120 \]

#### Step 4.4: Count words starting with 'CK'
If the word starts with 'CK', the remaining letters are C, E, I, R, T. The number of permutations of these letters is:
\[ 5! = 120 \]

#### Step 4.5: Count words starting with 'CR'
Now we need to consider words starting with 'CR'. The next letter can be E, I, K, or T.

1. **CR + E**: Remaining letters are C, I, K, T.
\[ 4! = 24 \]
2. **CR + I**: Remaining letters are C, E, K, T.
\[ 4! = 24 \]
3. **CR + K**: Remaining letters are C, E, I, T.
\[ 4! = 24 \]
4. **CR + T**: Remaining letters are C, E, I, K.
\[ 4! = 24 \]

### Step 5: Count words starting with 'CRIC'
Now we need to consider words starting with 'CRI'. The next letter can be C, E, K, or T.

1. **CRI + C**: Remaining letters are E, K, T.
\[ 3! = 6 \]
2. **CRI + E**: Remaining letters are C, K, T.
\[ 3! = 6 \]
3. **CRI + K**: Remaining letters are C, E, T.
\[ 3! = 6 \]
4. **CRI + T**: Remaining letters are C, E, K.
\[ 3! = 6 \]

### Step 6: Count words starting with 'CRIC'
Now we need to consider words starting with 'CRIC'. The next letter can be E, K, or T.

1. **CRIC + E**: Remaining letters are K, T.
\[ 2! = 2 \]
2. **CRIC + K**: Remaining letters are E, T.
\[ 2! = 2 \]
3. **CRIC + T**: Remaining letters are E, K.
\[ 2! = 2 \]

### Step 7: Total up all the counts
Now we can sum up all the counts:
- Words starting with 'CC': 120
- Words starting with 'CE': 120
- Words starting with 'CI': 120
- Words starting with 'CK': 120
- Words starting with 'CR' and E: 24
- Words starting with 'CR' and I: 24
- Words starting with 'CR' and K: 24
- Words starting with 'CR' and T: 24
- Words starting with 'CRI' and C: 6
- Words starting with 'CRI' and E: 6
- Words starting with 'CRI' and K: 6
- Words starting with 'CRI' and T: 6
- Words starting with 'CRIC' and E: 2
- Words starting with 'CRIC' and K: 2
- Words starting with 'CRIC' and T: 2

Adding these gives:
\[ 120 + 120 + 120 + 120 + 24 + 24 + 24 + 24 + 6 + 6 + 6 + 6 + 2 + 2 + 2 = 530 \]

### Final Answer
Thus, the number of words before the word "CRICKET" is **530**.