We are given information about students born in different months based on the letters in the month names, and we need to determine how many students were born in May.

Let's summarize the given information:

• Total students: $177$
• $63$ students were born in months with at least one 'u' (January, February, June, July, August).
• $112$ students were born in months with at least one 'r' (January, February, March, April, September, October, November, December).
• $71$ students were born in months with at least one 'm' and at least one 'r' (March, September, November, December).
• $23$ students were born in the first two months of the year (January and February).

Step 1: Identify overlap between months

We are asked to focus on students born in May, but the information is organized by various letters in the month names. Since we are only interested in May, we note that:

• May does not have a 'u', so it is not part of the 63 students born in months with at least one 'u'.
• May does not have an 'r', so it is not part of the 112 students born in months with at least one 'r'.
• May does not have both 'm' and 'r', so it is not part of the 71 students born in months with both letters.

Thus, no group described includes students born in May, and we can calculate the number of students born in May by subtracting the total number of students in all the described groups from the total number of students.

Step 2: Total students in all described groups

• The 63 students born in 'u' months and the 112 students born in 'r' months likely overlap.
• Also, there is overlap within the 71 students born in months with 'm' and 'r'.

However, the problem does not provide detailed overlap information between the various groups except for the first two months. So, we assume that all the students in the given groups collectively account for the entire student population except those born in May.

Thus, to find the number of students born in May, we subtract the total number of students in described groups (months with 'u', 'r', etc.) from the total number of students.

Step 3: Subtract students born in non-May months

If 63 students are born in months with 'u', 112 are born in months with 'r', and 23 are born in January and February, these likely describe overlapping sets. Therefore, we assume that all other students are distributed across the other months except May.

Thus, the number of students born in May is:

$\text{Students in May} = 177 - (63 + 112 + 71 + 23)$

This leads us to:

$\text{Students in May} = 177 - (63 + 112 + 71 + 23) = 177 - 269 = -92$

This suggests there is an issue with overlaps in groupings. We may need additional clarifications. Would you like me to continue with assumptions to recheck overlap handling?