Question

An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, ..., 9 such that the first digit of the code is non-zero. The code, handwritten on a slip, can however potentially create confusion when read upside down — for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?

• A. 80
• B. 71
• C. 78
• D. 69

Hint:

Only the digits 0, 1, 8 (which look the same upside down) and 6, 9 (which swap) can be misread. Count the codes made only of these digits that give a different valid code when turned around.

Answer 1

The Correct Option is B

Step 1: Total codes:
First digit non-zero (9 choices), second digit different from it (9 choices): \( 9 \times 9 = 81 \) codes.

Step 2: Which codes can cause confusion?:
Turning a slip upside down reverses the order of the digits and turns each digit around: 0 stays 0, 1 stays 1, 8 stays 8, 6 becomes 9 and 9 becomes 6. The other digits (2, 3, 4, 5, 7) do not give a digit, so they cannot be confused. A code is confusing when both digits are from {0, 1, 6, 8, 9}, the turned code is a valid code (it must not start with 0, so the second digit of the original is not 0) and it is a different code.

Step 3: Count the confusing codes:
First digit from {1, 6, 8, 9} and second digit from {1, 6, 8, 9} other than the first: \( 4 \times 3 = 12 \) codes: 16, 18, 19, 61, 68, 69, 81, 86, 89, 91, 96, 98. Among them 69 and 96 read the same upside down (69 reads as 69 and 96 reads as 96), so there is no confusion. That leaves 10 confusing codes: 16, 18, 19, 61, 68, 81, 86, 89, 91, 98. Codes ending in 0, like 10, turn into 01 which is not a valid code, so they are not confusing.

Step 4: Answer:
Codes with no confusion \( = 81 - 10 = 71 \).

Final Answer:
The correct option is (2): 71