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 nonzero. 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? 80 78 71 69

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 nonzero. 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?

• 80
• 78
• 71
• 69

The correct answer is D: 69

Consider the available digits: $$0, 1, 2, 3, \ldots, 9$$. For the code formation:

• The first digit must be nonzero, so it can be chosen in 9 ways (from 1 to 9).
• The second digit can be any digit except the first one, hence 9 ways (as there is no repetition allowed).

Thus, the total number of possible codes is:

$$ 9 \times 9 = 81 $$

Next, identify the digits that could cause confusion when read upside down. These digits are 1, 6, 8, and 9.

• If the first digit is one of these confusing digits (4 choices: 1, 6, 8, 9),
• The second digit can be the remaining three confusing digits (3 ways for each case).

Thus, the total number of confusing codes is:

$$ 4 \times 3 = 12 $$

Finally, calculate the count of codes where no confusion can arise by subtracting the confusing codes from the total possible codes:

$$ 81 - 12 = 69 $$

Therefore, the number of codes for which no confusion can arise is:

69