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 nPr.
→ 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 = 2 no of I = 2. Thus, n=11,p=2,q=2 so, 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 nCr.
Formula:
