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. 78
• C. 71
• D. 69

Hint:

When considering numbers that may create confusion when flipped, identify the digits that are symmetric and exclude invalid pairs.

Answer 1

The Correct Option is B

The digits that can create confusion when read upside down are: 0, 1, 6, 8, 9. Therefore, the digits that are allowed in the code without creating confusion are 1, 6, 8, 9, which can be read as themselves when flipped upside down. For the first digit, we can select from 1, 6, 8, 9 (4 choices). For the second digit, we can select from 0, 1, 6, 8, 9 (5 choices). Thus, the total number of valid codes is: \[ 4 \times 5 = 20 \] However, we need to exclude the invalid cases where the code reads as a different number when flipped. We can list these numbers: 69, 96, 18, 81, and 88. Thus, the total number of codes without confusion is: \[ 4 \times 5 - 5 = 78 \]