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

Combination

In a combination order is not important. We are selecting objects in different ways.

It is denoted by ^{n}C_{r} or C(n,r).

\rightarrow\ ^{n}C_{r}=\dfrac{n!}{r!\ (n-r)!}\ \ (r<n).

\rightarrow\ ^{n}P_{r}=r!\times ^{n}C_{r}.

\rightarrow\ ^{n}C_{0}=1.

\rightarrow\ ^{n}C_{1}= ^{n}C_{n-1}=n.

\rightarrow\ ^{n}C_{r}= ^{n}C_{n-r}.

\rightarrow If ^{n}C_{x}= ^{n}C_{y}\ \Rightarrow\ x=y\text{ or }x+y=n.

\rightarrow\ ^{n}C_{r}+ ^{n}C_{r-1}=^{n+1}C_{r}.

\rightarrow\ ^{n}C_{0}+^{n}C_{1}+\cdots+^{n}C_{n}=2^{n}.

\rightarrow\ ^{n}C_{1}+^{n}C_{1}+\cdots+^{n}C_{n}=2^{n}-1.

\rightarrow The number of non-negative solutions of the equation x_{1}+x_{2}+\cdots +x_{m}=n\ \ (n>m) is given by ^{n+m-1}C_{m-1} or ^{n+m-1}C_{n}.

\rightarrow No. of combinations when ‘p’ particular things are always included =^{n-p}C_{r}.

\rightarrow No. of ways of selecting zero or more things out of ‘n’ identical things is 1+1+\cdots (n+1\ \text{terms})=n+1.

\rightarrow Number of circular arrangement of n things is \dfrac{n!}{n}=(n-1)!.

\rightarrow Number of circular permutations of n dissimilar things taken r at a time

=\begin{cases} \dfrac{^{n}P_{r}}{r}, & \mbox{if clockwise and anticlockwise orders are considered as different}  \vspace*{0.18cm}\\ \dfrac{^{n}P_{r}}{2r}, & \mbox{if clockwise and anticlockwise orders are considered as same}  \end{cases}

\rightarrow The number of ways on which ‘n’ things can be distributed among two persons, one receives ‘p’ and another receives q thing s-t

(p+q-n)=^{n}C_{p}\cdot ^{n-p}C_{p}=\dfrac{n!}{p!\ q !}.

\rightarrow If we want to keep the things in two heaps then the number of selection =\dfrac{n!}{p!\ q !\ 2!}.

\rightarrow If we want to keep identical things in 3 heaps then no. of selection =\dfrac{(3n)!}{n!\ n!\ n!\ 3!}.

\rightarrow If 3n things are to be divided equally among three people, then the no. of ways =\dfrac{(3n)!}{(n!)^{3}}.

\rightarrow No. of ways in which (m+n+p) different things can be divided into three different groups containing m,n\ \&\ P respectively is \dfrac{(m+n+P)!}{m!\ n!\ P!}.

Dearrangements :

 No. of ways in which ‘n’ letters can be put in n corresponding envelopes s-t no letter goes to correct envelope is n!(1-\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{3!}+\cdots +(-1)^{n}\dfrac{1}{n!}).

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