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

Options

A. 80
B. 78
C. 71
D. 69

Answer

To solve the problem step by step, we need to determine how many two-digit codes can be formed using distinct digits from 0 to 9, with the first digit being non-zero, and ensuring that no confusion arises when the code is read upside down.

### Step 1: Identify the total number of codes
1. The first digit can be any digit from 1 to 9 (since it cannot be zero). This gives us 9 options for the first digit.
2. The second digit can be any digit from 0 to 9, except the digit chosen as the first digit. This gives us 9 options for the second digit (10 total digits minus the 1 digit already used).

**Total codes = 9 (choices for first digit) × 9 (choices for second digit) = 81 codes.**

### Step 2: Identify the digits that cause confusion
The digits that can create confusion when read upside down are:
- 0 ↔ 0 (not confusing)
- 1 ↔ 1 (confusing)
- 6 ↔ 9 (confusing)
- 8 ↔ 8 (not confusing)
- 9 ↔ 6 (confusing)

From the above, the digits that can confuse each other are:
- 1, 6, 8, 9

### Step 3: Calculate the number of confusing codes
1. **Confusing pairs**: The confusing pairs are:
- (1, 1)
- (6, 9) and (9, 6)
- (8, 8)

2. **Count the confusing combinations**:
- For the digits 1, 6, and 9:
- 1 can pair with 1 (1, 1) - Not allowed since digits must be distinct.
- 6 can pair with 9 (6, 9) and (9, 6) - 2 combinations.
- For digit 8:
- 8 can pair with 8 (8, 8) - Not allowed since digits must be distinct.

Thus, the only confusing combinations are:
- 16 ↔ 91
- 69 ↔ 96

So, we have 4 confusing codes: 16, 19, 69, and 96.

### Step 4: Calculate the number of non-confusing codes
To find the number of codes that do not create confusion, we subtract the number of confusing codes from the total number of codes.

**Non-confusing codes = Total codes - Confusing codes = 81 - 4 = 77.**

### Final Answer
The number of codes for which no confusion can arise is **77**.

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