Solution:

Given: M n P2 = 56, m-nP2 = 30

Here, we are given two equations, which we need to solve to find the values of m and n.

Step 1: Understanding the notation

- M n P2 represents the number of permutations of n objects taken from a set of M objects, where order matters.
- m-nP2 represents the number of permutations of 2 objects taken from a set of n-m objects, where order matters.

Step 2: Using the given equations

- From the first equation, M n P2 = 56, we can write:
- n! / (n-2)! = 56 (using the formula for permutations)
- Simplifying this, we get:
- n(n-1) = 112
- Solving for n, we get n = 8 or n = -14 (which is not a valid solution since n has to be positive)

- From the second equation, m-nP2 = 30, we can write:
- (n-m)! / (n-m-2)! = 30 (using the formula for permutations)
- Simplifying this, we get:
- (n-m)(n-m-1) = 60
- We can use the value of n obtained from the first equation to solve for m. Substituting n = 8, we get:
- (8-m)(7-m) = 60
- Solving this quadratic equation, we get m = 3 or m = 5

Step 3: Checking the solutions

- We have obtained two possible values for m, which we need to check with the first equation to see if they satisfy it. Substituting m = 3 and n = 8, we get:
- 8P2 = 28, which is not equal to 56, so m = 3 is not a valid solution.
- Substituting m = 5 and n = 8, we get:
- 8P2 = 56, which is equal to the given value in the first equation, so m = 5 is a valid solution.
- Therefore, the solutions are: m = 5 and n = 8.

Step 4: Final answer

- The values of m and n that satisfy the given equations are: m = 5 and n = 8.