A **permutation **is an arrangement of objects in a specific order.

• In other words, **permutations **refer to the different ways that a set of items can be rearranged while maintaining the distinctness and order of the items.

• Permutations are used to count the number of possible orders or sequences that a set of items can be placed in.

**For example,**The permutations of the letters “

**A**,” “

**B**,” and “

**C**” would include “ABC,” “ACB,” “BAC,” “BCA,” “CAB,” and “CBA.”

• **Note**: An ordered arrangement of r objects from a set of n objects is called as r-permutation of n objects. It is denoted by **P(n,r) **or ^{n}P_{r}.

→ **The formula to calculate the number of permutations is:**

**Q).** In a hostel, there are **7 doors**, In how many ways can a student enter the hostel and come out by different door?

**solution:**

A student can enter the hostel in 7 different ways.Since, he has to come out by a different doors, he can come out the hostel in (7-1=6 ways).So, by multiplication principle of counting the student can enter and exit the hostel in 7*6=42 different ways |

## → Permutation without Repetitions:

The total number of permutation of a set of **n **distinct objects taken **r **at a time is given by :

## → Permutation with Repetitions:

The permutation of **n **objects taken all at a time, when there are **P **objects of one kind , **q **objects of second kind, **r **objects are of a third kind, is:

**Q).** In how many ways can the letters of the word “**PROBABILITY**” be arranged?

solution:

In the word “PROBABILITY“, there are 11 letters in which,no. of B = 2no of I = 2.Thus, n=11,p=2,q=2so, the required number of permutations: = n! /p!q!=11! /2!2!=11! /4=9979200 |

## → Circular Permutation:

A **circular permutation** is an arrangement of objects in a circle rather than in a straight line.

• The number of ways of arranging **n **unlike objects in a ring when clockwise and anticlockwise arrangements are different is **(n-1)! **.

**Note**:

if the clockwise and anticlockwise arrangements are not distinct as in the case of necklace of beads and beads into a bracelet, then **the required number of arrangements in a circle will be:**

**Q). **In how many ways can the numbers on a clock face be arranged?

**solution**:

In a clock face there are 12 numbers. So they can be arranged in (12-1)!=11! ways |

## → Combination:

A **combination **is a selection of objects from a set without regard to the order of selection.

**• Combinations **refer to the different ways that a subset of items can be chosen from a larger set, without considering the arrangement or order of the selected items.

**• **It is denoted by **C(n,r)** or** ^{n}C_{r}.**

**Formula**: