Permutations & Combinations
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Important Formulas & Concepts
Study MaterialPermutations & Combinations
Permutations and Combinations is one of the most important chapters in Quantitative Aptitude and Higher Mathematics. It is widely used in Probability, Statistics, Arrangement Problems, Group Selection, and Logical Counting questions.
This chapter is frequently asked in SSC, Banking, Railway, CAT, NDA, CDS, Insurance, Defence, and various competitive examinations.
The biggest confusion in this chapter is understanding the difference between:
- Permutation → Arrangement
- Combination → Selection
Permutation = Order Matters
Combination = Order Does Not Matter
What is Permutation?
The various ways of arranging a given number of objects by taking some or all at a time are called Permutations.
In permutation, arrangement or order is important.
Permutation is denoted by:
nPr or P(n,r)
Permutation Formula
The number of permutations of n objects taken r at a time:
nPr = n! / (n-r)!
where:
- n = Total number of objects
- r = Number of objects selected
Understanding Permutation with Example
Arrange numbers 1, 2, and 3 by taking two at a time.
Possible arrangements:
12, 21, 13, 31, 23, 32
Total arrangements = 6
Here:
- 12 and 21 are different
- 13 and 31 are different
Therefore, order matters.
Permutation of Letters
All permutations made with letters a, b, c taking two at a time:
ab, ba, ac, ca, bc, cb
Permutation Taking All Objects
All permutations of a, b, c taking all at a time:
abc, acb, bac, bca, cab, cba
Important Permutation Formulas
1. Permutation of n Different Objects
If all objects are different:
n!
ways
2. Permutation Taken r at a Time
nPr = n! / (n-r)!
3. Circular Permutation
Number of ways to arrange n objects in a circle:
(n - 1)!
4. Permutation with Repetition Allowed
When repetition is allowed:
nr
where:
- n = Total objects
- r = Number of positions
5. Permutation of Similar Objects
If some objects are identical:
N! / (B1! × B2! × B3! ...)
where:
- N = Total objects
- B₁, B₂, B₃ = Similar objects
Restricted Permutation
1. Particular Objects Never Included
If k objects are never included:
(n-k)Pr
2. Particular Objects Always Included
If k objects are always included:
(n-k)C(r-k) × r!
What is Combination?
The different groups or selections formed by taking some or all objects are called Combinations.
In combination:
- Only selection matters
- Order does not matter
Combination is denoted by:
nCr or C(n,r)
Combination Formula
Number of combinations of n objects taken r at a time:
nCr = n! / [r!(n-r)!]
Relationship Between Permutation and Combination
nPr = nCr × r!
Important Properties of Combination
1. Symmetry Property
nCr = nCn-r
2. Special Cases
nC0 = nCn = 1
3. Another Important Formula
nC1 = n
Example: Volunteer Selection
Suppose there are:
- 12 boys
- 8 girls
Total students:
= 20
Number of ways to select 5 volunteers:
20C5
Quick Revision Formula Table
| Concept | Formula |
|---|---|
| Permutation | nPr = n!/(n-r)! |
| Combination | nCr = n!/[r!(n-r)!] |
| Circular Permutation | (n−1)! |
| Permutation with Repetition | nr |
| Permutation-Combination Relation | nPr = nCr × r! |
Permutations and Combinations form the foundation for Probability and advanced counting techniques. Strong understanding of formulas and concepts helps candidates solve competitive examination questions quickly and accurately.