Question

Codes used in vehicle identification consists of two distinct English alphabets followed by two distinct digits from 1 to 9. How many of them end with an even number.

• A. 10400
• B. 2600
• C. 20800
• D. 5200

Hint:

Break permutation problems into stages: first letters, then digits, and apply conditions step-by-step.

Answer 1

The Correct Option is C

Step 1: Understand the structure of the code.
Each code consists of:
- 2 distinct English alphabets
- followed by 2 distinct digits (from 1 to 9)

Step 2: Choose the alphabets.
Total alphabets = 26.
Number of ways to choose 2 distinct alphabets in order:
\[ {}^{26}P_2 = 26 \times 25. \]

Step 3: Condition for last digit.
The code must end with an even number.
Even digits from 1 to 9 are:
\[ 2,4,6,8 \Rightarrow 4 \text{ choices}. \]

Step 4: Choose the second last digit.
The two digits must be distinct.
So, after choosing last digit, the second last digit can be any of the remaining 8 digits.

Step 5: Count digit arrangements.
Total ways for digits:
\[ 8 \times 4. \]

Step 6: Multiply total possibilities.
\[ \text{Total codes} = (26 \times 25) \times (8 \times 4). \]
\[ = 650 \times 32. \]
\[ = 20800. \]

Step 7: Final conclusion.
Thus, total number of such codes is 20800.
Final Answer:
\[ \boxed{20800}. \]