A certain online form requires a 2-digit code for the day of the month to be entered into one of...

1. Translate the problem requirements: We need to find valid 2-digit day codes (01-31) whose transpose (reversing the digits) creates an invalid code. For example, 04 is valid but its transpose 40 is invalid since there's no 40th day in a month.
2. Identify the structure of valid codes: Determine what makes a code valid (represents days 1-31) and systematically organize them by their first digit.
3. Apply the transpose operation systematically: For each valid code, create its transpose by swapping digits and check if the result falls outside the valid range of 01-31.
4. Count codes with invalid transposes: Systematically go through each group and count those whose transposes are not valid day codes.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday language. We have 2-digit codes that represent days of the month, like 04 for the 4th day. The "transpose" just means we flip the digits - so 04 becomes 40.

A code is valid if it represents a real day of the month: 01, 02, 03, all the way up to 31. Anything outside this range (like 00, 32, 40, etc.) is not valid.

We need to find valid codes whose transpose creates an invalid code. For example, 04 is valid (4th day exists), but its transpose 40 is invalid (there's no 40th day in any month).

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Identify the structure of valid codes

Let's organize all valid codes by looking at what the first digit can be:

Codes starting with 0: 01, 02, 03, 04, 05, 06, 07, 08, 09 (9 codes)

Codes starting with 1: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 (10 codes)

Codes starting with 2: 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 (10 codes)

Codes starting with 3: 30, 31 (2 codes)

Total valid codes: 9 + 10 + 10 + 2 = 31 codes (which makes sense - days 1 through 31)

3. Apply the transpose operation systematically

Now let's flip each code and see which transposes are invalid. Remember, a transpose is invalid if it's less than 01 or greater than 31.

Codes starting with 0 (01-09):

01 → 10 (valid), 02 → 20 (valid), 03 → 30 (valid), 04 → 40 (invalid), 05 → 50 (invalid), 06 → 60 (invalid), 07 → 70 (invalid), 08 → 80 (invalid), 09 → 90 (invalid)

Invalid transposes: 6 codes (04, 05, 06, 07, 08, 09)

Codes starting with 1 (10-19):

10 → 01 (valid), 11 → 11 (valid), 12 → 21 (valid), 13 → 31 (valid), 14 → 41 (invalid), 15 → 51 (invalid), 16 → 61 (invalid), 17 → 71 (invalid), 18 → 81 (invalid), 19 → 91 (invalid)

Invalid transposes: 6 codes (14, 15, 16, 17, 18, 19)

Codes starting with 2 (20-29):

20 → 02 (valid), 21 → 12 (valid), 22 → 22 (valid), 23 → 32 (invalid), 24 → 42 (invalid), 25 → 52 (invalid), 26 → 62 (invalid), 27 → 72 (invalid), 28 → 82 (invalid), 29 → 92 (invalid)

Invalid transposes: 7 codes (23, 24, 25, 26, 27, 28, 29)

Codes starting with 3 (30-31):

30 → 03 (valid), 31 → 13 (valid)

Invalid transposes: 0 codes

Process Skill: CONSIDER ALL CASES - Systematically checking each group to avoid missing any codes

4. Count codes with invalid transposes

Let's add up all the codes whose transposes are invalid:

From 0X codes: 6 codes (04, 05, 06, 07, 08, 09)

From 1X codes: 6 codes (14, 15, 16, 17, 18, 19)

From 2X codes: 7 codes (23, 24, 25, 26, 27, 28, 29)

From 3X codes: 0 codes

Total: N = 6 + 6 + 7 + 0 = 19

Final Answer

The number of valid codes having a transpose that is not valid is N = 19.

This matches answer choice D. 19.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "transpose" means: Students might think transpose means something complex rather than simply swapping the two digits. For example, they might think 04 transpose is 4 (dropping the zero) rather than 40, leading to completely wrong calculations.

2. Confusion about the direction of the problem: Students may incorrectly try to find invalid codes whose transpose is valid, rather than valid codes whose transpose is invalid. This reversal would lead them to a completely different set of numbers to count.

3. Misinterpreting the valid range: Students might forget that valid codes are specifically 01 through 31 (not 00 through 31 or 1 through 31 without leading zeros), which would affect both their list of valid codes and their assessment of which transposes are invalid.

Errors while executing the approach

1. Systematic counting errors: When going through each group (0X, 1X, 2X, 3X), students often miscount or skip codes, especially in the longer lists. For example, they might count only 5 codes instead of 6 in the 04-09 group that have invalid transposes.

2. Incorrect transpose calculations: Students may make errors when mentally flipping digits, particularly with codes like 30→03 or 13→31, especially when working quickly through multiple cases.

3. Wrong validity assessments: Students might incorrectly classify a transpose as valid or invalid. For example, they might think 32 is valid (since it's close to 31) or forget that anything above 31 is automatically invalid.

Errors while selecting the answer

No likely faltering points - the final step is straightforward addition of clearly counted groups, and the answer choices are sufficiently spread apart that minor arithmetic errors would still lead to an obviously incorrect choice.