Permutations and Combinations Class 11 Notes Maths Chapter 7

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CBSE Class 11 Maths Notes Chapter 7 Permutations and Combinations

Permutation And Combination Class 11 Notes
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.

Permutations And Combinations Class 11 Notes Pdf
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

Class 11 Maths Chapter 7 Notes
Permutation
Each of the different arrangement which can be made by taking some or all of a number of objects is called permutation.

Permutations And Combinations Class 11 Notes
Permutation of n different objects
The number of arranging of n objects taking all at a time, denoted by ⁿP_n, is given by ⁿP_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
ⁿP_r = \(\frac { n! }{ \left( n-r \right) ! }\)

Properties of Permutation

Permutations And Combinations Notes Class 11
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 p₁ + p₂ + p₃ + … + 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-1P_r-1

When a particular object is always excluded, then number of arrangements = ^n-1P_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)!.

Permutation And Combination Notes Class 11
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 ⁿC_r, and is given by
ⁿC_r = \(\frac { n! }{ r!\left( n-r \right) ! }\)

Properties of Combinations

NCERT Solutions