The number of words of that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a consonant, is

Options

A. `(3)/(2)(7)!`
B. `2(7)!`
C. `(5)/(2)(7)!`
D. `3(7)!`

Answer

To solve the problem of finding the number of words that can be formed from the letters of the word "CALCULATE," such that each word starts and ends with a consonant, we can follow these steps:

### Step 1: Identify the consonants and vowels in the word "CALCULATE."
- The consonants are: C, L, C, L, T (5 consonants)
- The vowels are: A, U, A, E (4 vowels)

### Step 2: Determine the total number of letters.
The total number of letters in "CALCULATE" is 9.

### Step 3: Identify the possible consonants for the first and last positions.
The consonants that can be used to start and end the word are C, L, and T.

### Step 4: Calculate the cases based on different consonant combinations for the first and last letters.

#### Case 1: Starts with C and ends with C.
- The arrangement will look like: C _ _ _ _ _ _ _ C
- We have 7 positions left to fill with the remaining letters: L, L, T, A, U, A (total of 7 letters).
- The number of arrangements is given by:
\[ \frac{7!}{2! \cdot 2!} = \frac{5040}{4} = 1260 \]

#### Case 2: Starts with L and ends with L.
- The arrangement will look like: L _ _ _ _ _ _ _ L
- The remaining letters are: C, C, T, A, U, A (total of 7 letters).
- The number of arrangements is given by:
\[ \frac{7!}{2! \cdot 2!} = \frac{5040}{4} = 1260 \]

#### Case 3: Starts with C and ends with T.
- The arrangement will look like: C _ _ _ _ _ _ _ T
- The remaining letters are: L, L, C, A, U, A (total of 7 letters).
- The number of arrangements is given by:
\[ \frac{7!}{2! \cdot 2!} = \frac{5040}{4} = 1260 \]

#### Case 4: Starts with T and ends with C.
- The arrangement will look like: T _ _ _ _ _ _ _ C
- The remaining letters are: L, L, C, A, U, A (total of 7 letters).
- The number of arrangements is given by:
\[ \frac{7!}{2! \cdot 2!} = \frac{5040}{4} = 1260 \]

#### Case 5: Starts with T and ends with L.
- The arrangement will look like: T _ _ _ _ _ _ _ L
- The remaining letters are: C, C, L, A, U, A (total of 7 letters).
- The number of arrangements is given by:
\[ \frac{7!}{2! \cdot 2!} = \frac{5040}{4} = 1260 \]

### Step 5: Sum all the cases.
The total number of arrangements is:
\[ 1260 + 1260 + 1260 + 1260 + 1260 = 6300 \]

### Final Answer:
The total number of words that can be formed by writing down the letters of the word "CALCULATE," such that each word starts and ends with a consonant, is **6300**.