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A code word consists of three letters of...

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

A

1404000

B

16848000

C

2808000

D

none of these

Text Solution

AI Generated Solution

The correct Answer is:
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 into two parts: selecting the letters and selecting the digits. ### Step 1: Calculate the number of ways to choose the letters 1. **First Letter**: We can choose any of the 26 letters from the English alphabet. So, there are 26 options for the first letter. 2. **Second Letter**: Since we cannot repeat the letter chosen first, we have 25 options left 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: \[ \text{Ways to choose letters} = 26 \times 25 \times 24 \] ### Step 2: Calculate the number of ways to choose the digits 1. **First Digit**: We can choose any of the 10 digits (0-9). So, there are 10 options for the first digit. 2. **Second Digit**: Since we cannot repeat the digit chosen first, we have 9 options left for the second digit. The total number of ways to choose the digits is: \[ \text{Ways to choose digits} = 10 \times 9 \] ### Step 3: Combine the results To find the total number of code words, 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: Calculate the final result Now, we compute the values: 1. Calculate the number of ways to choose letters: \[ 26 \times 25 = 650 \] \[ 650 \times 24 = 15600 \] 2. Calculate the number of ways to choose digits: \[ 10 \times 9 = 90 \] 3. Finally, multiply the two results: \[ 15600 \times 90 = 1404000 \] Thus, the total number of code words is **1,404,000**. ### Final Answer The total number of code words is **1,404,000**. ---
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