A code word consists of three letters of the English alphabet followed by two digits of the decimal system. If neither letter nor digit is repeated in any code word, then the total number of code words, is

Options

A. 1404000
B. 16848000
C. 2808000
D. none of these

Answer

To find the total number of code words consisting of three letters followed by two digits, where neither letters nor digits are repeated, we can break down the problem step by step.

### Step 1: Choosing the Letters
1. **First Letter**: We can choose the first letter from the 26 letters of the English alphabet. So, there are 26 options.
2. **Second Letter**: Since the first letter cannot be repeated, we have 25 remaining letters to choose from for the second letter.
3. **Third Letter**: Similarly, for the third letter, we cannot use the first two letters, leaving us with 24 options.

The total number of ways to choose the letters is:
\[ 26 \times 25 \times 24 \]

### Step 2: Choosing the Digits
1. **First Digit**: We can choose the first digit from the 10 digits (0-9). So, there are 10 options.
2. **Second Digit**: Since the first digit cannot be repeated, we have 9 remaining digits to choose from for the second digit.

The total number of ways to choose the digits is:
\[ 10 \times 9 \]

### Step 3: Calculating the Total Code Words
Now, we multiply the number of ways to choose the letters by the number of ways to choose the digits:
\[ \text{Total Code Words} = (26 \times 25 \times 24) \times (10 \times 9) \]

### Step 4: Performing the Calculation
First, calculate the letters:
\[ 26 \times 25 = 650 \]
\[ 650 \times 24 = 15600 \]

Now, calculate the digits:
\[ 10 \times 9 = 90 \]

Finally, multiply the results:
\[ 15600 \times 90 = 1404000 \]

### Conclusion
Thus, the total number of code words is:
\[ \text{Total Code Words} = 1404000 \]

### Final Answer
The total number of code words is **1,404,000**.

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