Problem:
The average marks per student of classes A and B put together is 71. The average marks per student of classes B and C put together is 76. The average marks per student of classes A and C put together is 79. Find the range of the average marks (P) of all three classes put together.

Solution:
Let's assume the number of students in classes A, B, and C as A, B, and C respectively.

Average marks of classes A and B:
The average marks per student of classes A and B put together is 71. Therefore, the total marks of classes A and B put together = 71 * (A + B).

Average marks of classes B and C:
The average marks per student of classes B and C put together is 76. Therefore, the total marks of classes B and C put together = 76 * (B + C).

Average marks of classes A and C:
The average marks per student of classes A and C put together is 79. Therefore, the total marks of classes A and C put together = 79 * (A + C).

Total marks of all three classes:
The total marks of all three classes = (71 * (A + B)) + (76 * (B + C)) + (79 * (A + C)).

Total number of students:
The total number of students = A + B + C.

Range of average marks (P):
The average marks (P) of all three classes put together = Total marks of all three classes / Total number of students.

We need to find the range of P, which is the difference between the maximum and minimum possible values of P.

To find the maximum value of P, we need to maximize the total marks and minimize the total number of students.
To find the minimum value of P, we need to minimize the total marks and maximize the total number of students.

Maximum Value of P:
We can maximize the total marks by maximizing the values of A, B, and C.
Let's assume A, B, and C as positive integers.
In this case, the maximum value of P will occur when A = B = C.
Let's assume A = B = C = X.
The maximum value of P = (71 * (X + X)) + (76 * (X + X)) + (79 * (X + X)) / (X + X + X).

Simplifying the above expression:
Maximum value of P = 226(X) / 3.

Minimum Value of P:
We can minimize the total marks by minimizing the values of A, B, and C.
Let's assume A, B, and C as positive integers.
In this case, the minimum value of P will occur when one of the variables is 0 and the other two variables are positive integers. Let's assume A = 0, B = Y, and C = Z.
The minimum value of P = (71 * (0 + Y)) + (76 * (Y + Z)) + (79 * (0 + Z)) / (0 + Y + Z).

Simplifying the above expression:
Minimum value of P = (76 * (Y + Z)) / (Y + Z).