Permutations with restrictions

Primarily, there are two types of restrictions,

1. When one object is always included in each and every permutations
2. When one object is always excluded from all permutations.

For example, if there are 10 objects taken 3 at a time, then either a particular object will always be one of the 3 objects or that object will never be chosen as a member of the group of 3 objects. We will discuss each type of restriction here,

1. When an object is always included

Suppose there are n objects from which r objects are taken at a time. One object is always included in each permutation. Thus, the set of 'r' number of objects obtained as each permutation will always contain the one particular specified object.

Since one object is always included in every permutation, the remaining n-1 objects are to be arranged by taking r-1 objects at a time. Total number of such permutations is represented by P(n-1, r-1).

On the other hand, the object that is always included can be arranged in r different places in each permutation obtained above. Hence the total number of permutations is 

r * P(n-1, r-1)

2. When an object is always excluded

Since one object is always excluded from each permutation, hence, in a way, you have to calculate the permutations of n-1 objects taken r at a time. This is represented by P(n-1, r). Hence when an object is always excluded, the number of permutations is

P(n-1, r)