Permutations: Concepts, Principles & Examples
Permutation: The ways of arranging or selecting a smaller or equal number of persons or objects from a group of persons or objects with regard to the order of arrangement or selection, are called permutations.
In simple words, it means arrangement of things where the order of things is considered.
Example 1: Forming 3-digit numbers
Let us consider an example: We have to form a number consisting of three digits using the digits 1, 2, 3. The total number of possible arrangements will be 6 (Factorial 3 i.e., 3 × 2 × 1):
- Alternative 1: (1, 2, 3)
- Alternative 2: (2, 3, 1)
- Alternative 3: (3, 1, 2)
- Alternative 4: (1, 3, 2)
- Alternative 5: (2, 1, 3)
- Alternative 6: (3, 2, 1)
Each one of these possibilities is called a permutation of three digits taken all at a time.
nPr = n (n - 1) (n - 2) ... (n - r + 1)
nPr = n! / (n - r)!
Example 2: Forming 2-digit numbers
Again, if we have to form a number consisting of two digits using the digits 1, 2, 3, the total number of possible arrangements following the above formula will be 6:
- Alternative 1: (1, 2)
- Alternative 2: (2, 3)
- Alternative 3: (3, 1)
- Alternative 4: (1, 3)
- Alternative 5: (2, 1)
- Alternative 6: (3, 2)
Each one of these possibilities is called a permutation of two digits taken at a time.
Fundamental Principles of Counting
(a) Multiplication Rule: (AND = Multiply)
If a certain thing can be done in ‘m’ different ways and when it has been done, a second thing can be done in ‘n’ different ways, then the total number of ways of doing both things simultaneously is equal to m × n.
E.g., If one can go to school by 5 different buses and then come back by 4 different buses, then total number of ways of going to and coming back from school = 5 × 4 = 20 ways.
(b) Addition Rule: (OR = Add)
If there are two different jobs which can be done in ‘m’ ways and in ‘n’ ways respectively, then either of two jobs can be done in (m + n) ways.
E.g., If one wants to go school either by 5 buses or by auto where there are 4 autos, then total number of ways of going = 5 + 4 = 9 ways.
Practical Approach
A) Permutation of things when they all are different
To find the total number of permutation of n different things taken n or r at a time will be: nPr = n! / (n - r)!
6P6 = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways
6P3 = 6 × 5 × 4 = 120 ways
B) Permutation of things when they all are not different
To find the total number of permutation of n things taken all at a time where items repeat: Total = n! / [(p)! (q)! (r)!]
10! / [(2)! (2)!] = 9,07,200 ways
C) Permutation of things which may be repeated
To find the total number of permutation of n different things in which any item can be repeated without restriction, the total number of possible arrangements will be nr.
Ex 1. In a quiz competition there are 6 students. In how many ways 1st, 2nd, and 3rd prizes can be awarded to 6 students? Again, if there are three prizes—one in quiz, one in sport, and one in drawing—in how many ways can these prizes be distributed?
1st case: Since prizes are based on position (no repetition):
• 1st prize can be awarded in 6 ways
• 2nd prize can be awarded in 5 ways
• 3rd prize can be awarded in 4 ways
Total no. of permutations (nPr) = 120 ways
2nd case: Since prizes are based on competition type, all prizes can be awarded to the same student:
• 1st prize can be awarded in 6 ways to 6 students
• 2nd prize can be awarded in 6 ways to 6 students
• 3rd prize can be awarded in 6 ways to 6 students
Total no. of permutations (nr) = 63 = 216 ways
D) Permutation in a ring or in a circle
To find the total number of circular permutations of n different items, the formula is (n - 1)!.
Ex 1. In how many ways can 6 persons be arranged at a round table so that 2 particular persons may sit together?
• Arrangement with respect to table: Taking 2 persons as 1 person, we arrange 5 (=1+4) persons in 5! ways. Again, 2 persons may sit amongst themselves in 2! ways.
Required arrangements = 5! × 2! = 240 ways
• Arrangement with respect to each other: At first, 2 particular persons can arrange themselves in 2! ways. Keeping them fixed and taking as 1 person, all 5 persons can be arranged in (5-1) = 4! ways.
Required arrangements = 4! × 2! = 48 ways
• Clockwise vs Anti-clockwise: Suppose we arrange 6 persons such that no person has the same neighbours in both clockwise and anti-clockwise directions:
Required permutations = ½ × (6 - 1)! = 60 ways
E) Restricted Permutations
Ans: n = 10, r = 3, p = 2
Number of ways =
10 - 2P3 = 8P3 = 336 ways
Ans: n = 5, r = 3, p = 1
Number of ways =
5 - 1P3 - 1 = 4P2 = 12 ways
Ans: n = 10, r = 3, p = 2
Number of ways =
10 - 2P3 - 2 × 3P2 = 8P1 × 3P2 = 8 × 3 = 24 ways
Ans: n = 5, r = 3, p = 2
Number of ways =
(3 - 2 + 1) × 5 - 2P3 - 2 = (2) × 3P1 = 2 × 3 = 6 ways
12 × 2! = 24 ways (where 2! accounts for internal arrangement of 1 and 2).
Ans:
3! / 2! = 3 ways(Permutations are: NOT, OTN, ONT)
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