Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Permutation

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

\rightarrow The number of permutations of ‘n’ different objects taken ‘r’ at a time, where 0<r\leq n and the objects do not repeat in n(n-1)(n-2)\cdots (n-r+1) which is denoted by ^{n}P_{r}

i.e ^{n}P_{r}=n(n-1)(n-2)\cdots (n-r+1)\ \Rightarrow\ ^{n}P_{r}=\dfrac{n!}{(n-r)!}.

\rightarrow Number of permutation of ‘n’ distinct things take all at a time =n!.

\begin{array}{ll}  \rightarrow  & 0!=1 \\  & 1!=1\\  & 2!=1\times 2=2\\  & 3!=1\times 2\times 3=6\\  &4!=1\times 2\times 3\times 4=24\\  &\vdots\\  &n!=1\times 2\times 3\times 4\cdots\times (n-1)\times n\\  \end{array}

\rightarrow ^{n} P_{r} is also denoted as P(n,r).

\rightarrow Number of permutation of ‘n’ distinct things taken ‘r’ at a time when repetition is allowed =n^{r}.

\rightarrow Number of permutations of ‘n’ objects where p are of one kind, ‘q’ of second kind  and ‘r’ are of 3^{\text{rd}} kind s-t n-(p+q+r) i.e \dfrac{n!}{p!\ q!\ r!}.

\rightarrow Number of circular permutation of ‘n’ objects taken all at a time =(n-1)!

\rightarrow Number of circular permutation of ‘n’ objects taken ‘r’ at a time

=\dfrac{^{n}P_{r}}{r}.

\rightarrow If clockwise and anticklockwise arrangements are identical then the number of circular permutation of ‘n’ objects taken all at a time =\dfrac{1}{2}(n-1)!.

\rightarrow No. of permutations when particular thing is included =r\cdot    {^n-1}P{r-1}.

\rightarrow No. of permutations when particular thing is not included =^{n-1}P_{r}.

\rightarrow ^{n}P_{r}=r\cdot^{n-1}P_{r-1}+^{n-1}P_{r}

\rightarrow\ ^{n}P_{n}=n!=n(n-1)!.

\rightarrow\ (2n)!=2^{n}\cdot n!.

\rightarrow Factorials of negative intergrs are not defined.

\rightarrow\ ^{n}P_{1}=n.

\rightarrow No. of permutation when ‘r’ particular things are together =(n-r+1)!\ r!.

\rightarrow No. of permutation when ‘r’ particular things are identical =(n-r+1)!.

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