Permutation: Detailed Explanation | Mathematics (Maths) Class 11 - Commerce PDF Download

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

Properties of Permutation

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 p₁ are 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)!.

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].

(iii) The number of ways of dividing 2n different elements into two groups of n object when no distinction can be made between the groups i.e., order of the group is not important is [(2n)!/2!(n!)²].

This can be extended to 3n different elements into 3 groups is [(3n)!/3!(n!)³].

The number of ways in which mn different things can be divided equally it into m groups, if order of the group is not important is [(mn)!/(n!)^m m!].

(v) If the order of the group is important, then number of ways of dividing mn different things equally into m distinct groups is mn [(mn)!/(n!)^m]

(vi) The number of ways of dividing n different things into r groups is [rⁿ — ^rC₁(r — 1)ⁿ + ^rC₂(r — 2)ⁿ — ^rC₃(r – 3)ⁿ + …].

(vii) The number of ways of dividing n different things into r groups taking into account the order of the groups and also the order of things in each group is ^n+r-1P_n = r(r + l)(r + 2) … (r + n – 1).

(viii) The number of ways of dividing n identical things among r persons such that each gets 1, 2, 3, … or k things is the coefficient of x^{n – r} in the expansion of (1 + x + x² + … + X^k-1)^r.

Circular Permutation

In a circular permutation, firstly we fix the position of one of the objects and then arrange the other objects in all possible ways.

(i) Number of circular permutations at a time is (n -1)!. If clockwise taken as different. of n and different things taken anti-clockwise orders all are

(ii) Number of circular permutations of n different things taken all at a time, when clockwise or anti-clockwise order is not different 1/2(n – 1)!.

(iii) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are take as different is ⁿP_r/r

(iv) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are not different is ⁿP_r/2r.

(v) If we mark numbers 1 to n on chairs in a round table, then n persons sitting around table is n!.