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Maths class 11 Chapter 7. Permutations and
Combinations
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 npr
Properties of Permutation
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Important Results on’Permutation
1. The number of permutations of n different things taken r at a time, allowing repetitions
is nr.
2. The number of permutations of n different things taken all at a time is nPn= 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 p1 are alike of one kind p2 are alike of
second kind, p3 are alike of third kind,…, Pr are alike of rth kind such that p1 + p2 +
p3 +…+pr = n is n!/P1!P2!P3!….Pr!
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 – 1Pr – 1.
when a particular thing is always excluded, then number of arrangements = n – 1Pr
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)!.
Division into Groups
(i) The number of ways in which (m + n) different things can be divided into two groups which
contain m and n things respectively [(m + n)!/m ! n !].
This can be extended to (m + n + p) different things divided into three groups of m, n, p things
respectively [(m + n + p)!/m!n! p!].
(ii) The number of ways of dividing 2n different elements into two groups of n objects each is
[(2n)!/(n!)2] , when the distinction can be made between the groups, i.e., if the order of group is
important. This can be extended to 3n different elements into 3 groups is [(3n)!/((n!)3].
Maths class 11 Chapter 7. Permutations and
Combinations
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 npr
Properties of Permutation
, 2|Page
Important Results on’Permutation
1. The number of permutations of n different things taken r at a time, allowing repetitions
is nr.
2. The number of permutations of n different things taken all at a time is nPn= 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 p1 are alike of one kind p2 are alike of
second kind, p3 are alike of third kind,…, Pr are alike of rth kind such that p1 + p2 +
p3 +…+pr = n is n!/P1!P2!P3!….Pr!
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 – 1Pr – 1.
when a particular thing is always excluded, then number of arrangements = n – 1Pr
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)!.
Division into Groups
(i) The number of ways in which (m + n) different things can be divided into two groups which
contain m and n things respectively [(m + n)!/m ! n !].
This can be extended to (m + n + p) different things divided into three groups of m, n, p things
respectively [(m + n + p)!/m!n! p!].
(ii) The number of ways of dividing 2n different elements into two groups of n objects each is
[(2n)!/(n!)2] , when the distinction can be made between the groups, i.e., if the order of group is
important. This can be extended to 3n different elements into 3 groups is [(3n)!/((n!)3].
