## CBSE Class 11 Maths Notes Chapter 7 Permutations and Combinations

**Fundamental Principles of Counting**

Multiplication Principle: Suppose an operation A can be performed in m ways and associated with each way of performing of A, another operation B can be performed in n ways, then total number of performance of two operations in the given order is mxn ways. This can be extended to any finite number of operations.

**Addition Principle:** If an operation A can be performed in m ways and another operation S, which is independent of A, can be performed in n ways, then A and B can performed in (m + n) ways. This can be extended to any finite number of exclusive events.

**Factorial**

The continued product of first n natural number is called factorial ‘n’.

It is denoted by n! or n! = n(n – 1)(n – 2)… 3 × 2 × 1 and 0! = 1! = 1

**Permutation**

Each of the different arrangement which can be made by taking some or all of a number of objects is called permutation.

**Permutation of n different objects**

The number of arranging of n objects taking all at a time, denoted by ^{n}P_{n}, is given by ^{n}P_{n} = n!

The number of an arrangement of n objects taken r at a time, where 0 < r ≤ n, denoted by nP_{r} is given by

^{n}P_{r} = \(\frac { n! }{ \left( n-r \right) ! }\)

**Properties of Permutation**

**Important Results on Permutation**

The number of permutation of n things taken r at a time, when repetition of object is allowed is nr.

The number of permutation of n objects of which p1 are of one kind, p2 are of second kind,… pk are of kth kind such that p_{1} + p_{2} + p_{3} + … + p_{k} = n is \(\frac { n! }{ { { p }_{ 1 }!{ { p }_{ 2 }!{ p }_{ 3 }!…..{ p }_{ k }! } } }\)

Number of permutation of n different objects taken r at a time,

When a particular object is to be included in each arrangement is r. ^{n-1}P_{r-1}

When a particular object is always excluded, then number of arrangements = ^{n-1}P_{r}.

Number of permutations of n different objects taken all at a time when m specified objects always come together is m! (n – m + 1)!.

Number of permutation of n different objects taken all at a time when m specified objects never come together is n! – m! (n – m + 1)!.

**Combinations**

Each of the different selections made by taking some or all of a number of objects irrespective of their arrangements is called combinations. The number of selection of r objects from; the given n objects is denoted by ^{n}C_{r}, and is given by

^{n}C_{r} = \(\frac { n! }{ r!\left( n-r \right) ! }\)

**Properties of Combinations**

### CBSE Class 11 Maths Notes Chapterwise

- Chapter 1 Sets Class 11 Notes
- Chapter 2 Relations and Functions Class 11 Notes
- Chapter 3 Trigonometric Functions Class 11 Notes
- Chapter 4 Principle of Mathematical Induction Class 11 Notes
- Chapter 5 Complex Numbers and Quadratic Equations Class 11 Notes
- Chapter 6 Linear Inequalities Class 11 Notes
- Chapter 7 Permutations and Combinations Class 11 Notes
- Chapter 8 Binomial Theorem Class 11 Notes
- Chapter 9 Sequences and Series Class 11 Notes
- Chapter 10 Straight Lines Class 11 Notes
- Chapter 11 Conic Sections Class 11 Notes
- Chapter 12 Introduction to Three Dimensional Geometry Class 11 Notes
- Chapter 13 Limits and Derivatives Class 11 Notes
- Chapter 14 Mathematical Reasoning Class 11 Notes
- Chapter 15 Statistics Class 11 Notes
- Chapter 16 Probability Class 11 Notes