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).**

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