Number of Possible Arrangements of N Objects

  Recently I read a book on Probability and I have come across a few useful formulas to find out the number of possible arrangements of N objects with different restrictions. This type of questions are frequently asked in many competitive examinations held for persons with a Computer Science back ground. So I am posting them here.

• N distinct objects are arranged taking R (0<=R<=N) of them at a time with the following restrictions. Repetition is not allowed such that the same object X can't be used more than once. The order of the arrangement is relevant such that AB and BA are distinct. The total number of arrangements is given by the Permutation  P(N,R).

                                         P(N,R) = (N!)/((N-R)!).
                                 Where N! is the Factorial of N. 

• N distinct objects are arranged taking R (0<=R<=N) of them at a time with the following restrictions. Repetition is allowed such that the same object X can be used any number of times. The order of arrangement is relevant such that AB and BA are distinct. The total number of arrangements is given by the following formula.
  

                                         Number of Arrangements = N^R.
                                         Where N, R >= 0. 

• N distinct objects are arranged taking R of them at a time with the following restrictions. Repetition is not allowed such that the same object X can't be used more than once. The order of arrangement is immaterial such that AB and BA are the same. The total number of arrangements is given by Combination  C(N,R).
  

                                         C(N,R) = (N!)/((N-R)!(R!)).
                                         Where N! is the Factorial of N and N, R >= 0.

• N distinct objects are arranged taking R of them at a time with the following restrictions. Repetition is allowed such that the same object X can be used any number of times. The order of arrangement is immaterial such that AB and BA are the same. The total number of arrangements is given by Combination with repetition.
  

                      Number of Arrangements =  ((N+R-1)!)/((N-1)!(R)!).
                      Where N! is the Factorial of N and N, R >= 0.  

• Now consider the Circular Permutation of N distinct objects. The total number of arrangements is given by the following formula.
  

                                         Number of Arrangements = (N-1)!.
                                         Where (N-1)! is the Factorial of (N-1).