Permutations and Combinations: Maths Class 11 Chapter-7

Key Features of NCERT Material for Class 11 Maths Chapter 7 – Permutations and Combinations

Quick revision notes

In Chapter 6 of Class 11 : you must have learnt about Linear Inequalities. In chapter 7:  you will learn about Permutation and Combination.

Fundamental Principles of Counting

1. Multiplication Principle

If first operation can be performed in m ways and then a second operation can be performed in n ways. Then, the two operations taken together can be performed in mn ways. This can be extended to any finite number of operations.

2. Addition Principle

If first operation can be performed in m ways and another operation, which is independent of the first, can be performed in n ways. Then, either of the two operations can be performed in m + n ways. This can be extended to any finite number of exclusive events.

Factorial

For any natural number n, we define factorial as n ! or n = n(n – 1)(n – 2) … 3 x 2 x 1 and 0!= 1!= 1

Permutation

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

Mathematically  The number of ways of arranging n distinct objects in a row taking r (0 ≤ r ≤ n) at a time is denoted by P(n ,r) or ⁿp_r

Important Results on’Permutation

1. The number of permutations of n different things taken r at a time, allowing repetitions is n^r.
2. The number of permutations of n different things taken all at a time is ⁿP_n= n! .
3. The number of permutations of n things taken all at a time, in which p are alike of one kind, q are alike of second kind and r are alike of third kind and rest are different is n!/(p!q!r!)
4. The number of permutations of n things of which p₁are alike of one kind p₂ are alike of second kind, p₃ are alike of third kind,…, P_r are alike of rth kind such that p₁ + p₂ + p₃ +…+p_r = n is n!/P₁!P₂!P₃!….P_r!
5. Number of permutations of n different things taken r at a time,
   when a particular thing is to be included in each arrangement is r.^{n – 1}P_{r – 1}.
   when a particular thing is always excluded, then number of arrangements = ^{n – 1}P_r
6. Number of permutations of n different things taken all at a time, when m specified things always come together is m!(n – m + 1)!.
7. Number of permutations of n different things taken all at a time, when m specified things never come together is n! – m! x (n – m + 1)!.

Fundamental Principles of Counting

Multiplication Principle: Suppose an operation A can be performed in m ways and related with every method of performing of An, another operation B can be performed in n ways, at that point complete number of performance of two operations in the provided request is mxn ways. This can be stretched out to any limited 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, at that point An and B can performed in (m + n) ways. This can be stretched out to any limited number of restrictive events. 

Factorial 

The proceeded with result of first n common number is called factorial ‘n’. 

It is signified by n! or on the other hand n! = n(n – 1)(n – 2)… 3 × 2 × 1 and 0! = 1! = 1 

Permutation 

Every one of the distinctive arrangement which can be made by taking a few or the entirety of various items is called permutation. 

Permutation of n various articles 

The quantity of orchestrating of n objects taking all at once, signified by nPn, is given by nPn = n! 

The quantity of an arrangement of n objects taken r at once, where 0 < r ≤ n, meant by nPr is given by 

nPr =

Properties of Permutation

Important Results on Permutation

The quantity of permutation of n things taken r at once, when redundancy of article is permitted is nr. 

The quantity of permutation of n objects of which p1 are of one kind, p2 are of second kind,… pk are of kth kind with the end goal that p1 + p2 + p3 + … + pk = n is

Number of permutation of n various items taken r at once, 

At the point when a specific article is to be remembered for every arrangement is r. n-1Pr-1 

At the point when a specific article is constantly rejected, at that point number of arrangements = n-1Pr. 

Number of permutations of n various articles taken all when m indicated protests consistently meet up is m! (n – m + 1)!. 

Number of permutation of n various items taken all when m indicated protests never meet up is n! – m! (n – m + 1)!. 

Combinations 

Every one of the various determinations made by taking a few or the entirety of various items irrespective of their arrangements is called combinations. The quantity of choice of r objects from; the given n objects is indicated by nCr, and is given by 

nCr =

Properties of Combinations