Permutation And Combination
Arrangement:
$\quad$ Number of permutations of $n$ different things taken $r$ at a time $=$
$${ }^{n} P_{r}=n(n1)(n2) \ldots(nr+1)=\frac{n !}{(nr) !}$$
Circular permutation :
$\quad$ The number of circular permutations of $n$ different things taken all at a time is; $(n1)$ !
Selection :
$\quad$ Number of combinations of $n$ different things taken $r$ at a time
$$={ }^{n} C_{r}=\frac{n !}{r !(nr) !}=\frac{{ }^{n} P_{r}}{r !}$$
$\quad$ The number of permutations of ’ $n$ ’ things, taken all at a time, when ’ $p$ ’ of them are similar & of one type, $q$ of them are similar & of another type,
$\quad$ ’ $r$ ’ of them are similar & of a third type & the remaining $n(p+q+r)$ are all different is
$$\frac{n !}{p ! q ! r !}$$
Selection of one or more objects:
 Number of ways in which atleast one object be selected out of ’ $n$ ’ distinct objects is
$$ { }^{n} C_{1}+{ }^{n} C_{2}+{ }^{n} C_{3}+\ldots \ldots \ldots \ldots \ldots+{ }^{n} C_{n}=2^{n}1 $$
 Number of ways in which atleast one object may be selected out of ’ $p$ ’ alike objects of one type ’ $q$ ’ alike objects of second type and ’ $r$ ’ alike of third type is
$$ (p+1)(q+1)(r+1)1 $$
 Number of ways in which atleast one object may be selected from ’ $n$ ’ objects where ’ $p$ ’ alike of one type ’ $q$ ’ alike of second type and ’ $r$ ’ alike of third type and rest $n(p+q+r)$ are different is
$$(p+1)(q+1)(r+1) 2^{n(p+q+r)}1 $$
Multinomial theorem :
$\quad$ Coefficient of $X^{r}$ in expansion of
$$(1x)^{n}={ }^{n+r1} C_{r}(n \in N)$$
$\quad$ Let $N=p^{a} q^{b} r^{c} \ldots .$. where $p, q, r \ldots \ldots$ are distinct primes & $a, b, c \ldots .$. are natural numbers then :

The total numbers of divisors of $\mathrm{N}$ including 1 & N is $=(\mathrm{a}+1)(\mathrm{b}+1)(\mathrm{c}+1) \ldots \ldots \ldots$

The sum of these divisors is $=$
$$ \left(p^{0}+p^{1}+p^{2}+\ldots .+p^{a}\right)\left(q^{0}+q^{1}+q^{2}+\ldots .+q^{b}\right) \left(r^{0}+r^{1}+r^{2}+\ldots .+r^{c}\right) \ldots \ldots $$

Number of ways in which $\mathrm{N}$ can be resolved as a product of two factors is

$$ =\frac{1}{2}(a+1)(b+1)(c+1) \ldots . \hspace{2mm} \text { if N is not a perfect square.} $$

$$ =\frac{1}{2}[(a+1)(b+1)(c+1) \ldots+1] \hspace{2mm} \text { if N is a perfect square.}$$


Number of ways in which a composite number $\mathrm{N}$ can be resolved into two factors which are relatively prime (or coprime) to each other is equal to $2^{n1}$ where $\mathrm{n}$ is the number of different prime factors in $\mathrm{N}$.
Dearrangement :
$\quad$ Number of ways in which ’ $n$ ’ letters can be put in ’ $n$ ’ corresponding envelopes such that no letter goes to correct envelope is $$n !\left(1\frac{1}{1 !}+\frac{1}{2 !}\frac{1}{3 !}+\frac{1}{4 !} \ldots \ldots \ldots .+(1)^{n} \frac{1}{n !}\right)$$
Principle of counting :
PYQ2023P&CQ9, PYQ2023P&CQ12, PYQ2023P&CQ15, PYQ2023P&CQ20

The rule of sum: If a first task can be performed in m ways, while a second task can be performed in n ways, and the tasks cannot be performed simultaneously, then performing either one of these tasks can be accomplished in any one of total m+n ways.

The rule of product: If a procedure can be broken down into first and second stages, and if there are m possible outcomes for the first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in a total of mn ways.
Factorial notation:
$$n! = n(n1)…(3)(2)(1)$$
Permutation:
PYQ2023P&CQ1 , PYQ2023P&CQ8, PYQ2023P&CQ16, PYQ2023P&CQ20
The number of permutations of n different objects taken r at a time is represented as $$ ^n P_r = \frac{n!}{(nr)!}$$
 Permutation with repetition: PYQ2023P&CQ12
The no. of permutations of n different objects taken r at a time when each object may be repeated any number of times is $$n^r$$

Permutation of alike objects: The number of permutations of n objects taken all at a time in which, p are alike objects of one kind, q are alike objects of second kind & r are alike objects of a third kind and the rest (n  (p + q + r)) are all different is $$\frac{n!}{p! q! r!}$$

Permutation with Restriction: PYQ2023P&CQ7 PYQ2023P&CQ18
The number of permutations of n different objects, taken all at a time, when m specified objects always come together is $$m! × (n  m + 1)!$$
 Nonconsecutive selection: PYQ2023P&CQ10 PYQ2023P&CQ11 PYQ2023P&CQ14
$\quad \quad$• The number of selections of r consecutive objects out of n objects in a row $$= n  r + 1$$
$\quad \quad$• The number of selections of r consecutive objects out of n objects along a circle $$= n , \text{~ when~} r<n$$ $$= 1 , \text{~ when~} r=n$$
Circular permutation:
The number of circular arrangements of n different objects $$= (n  1)!$$

Clockwise and anticlockwise arrangements:

When clockwise and anticlockwise arrangements are not different, i.e. when observations can be made from both sides, the number of circular arrangements of n different objects is $$\frac{(n  1)!}{2}$$

The number of circular permutation of n different objects taken all at a time is (n  1)!, if clockwise and anticlockwise orders are taken as different.

The number of circular permutations of n different objects taken r at a time

$\frac{^nP_r}{r}$, when clockwise and anticlockwise orders are treated as different.

$\frac{^nP_r}{2r}$, when clockwise and anticlockwise orders are treated as same.


Properties of permutation:

${ }^n P_n=n(n1)(n2) \ldots 3 \times 2 \times 1=n$!

${ }^n P_0=\frac{n !}{n !}=1$

${ }^n P_1=n$

${ }^n P_{n1}=n$ !

${ }^n P_r=n \cdot{ }^{n1} P_{r1}=n(n1)^{n2} P_{r2}$

${ }^{n1} P_r+r \cdot{ }^{n1} P_{r1}={ }^n P_r$

$\frac{{ }^n P_r}{{ }^n P_{r1}}=nr+1$
Combination:
$\quad$ The number of all combinations of n objects, taken r at a time is generally denoted by C(n, r) or $^nC_r$ is $$ ^n C_r = \frac{n!}{r! (nr)!}$$

Properties:

${ }^n C_r={ }^n C_{nr}$

${ }^n C_r+{ }^n C_{r1}={ }^{n+1} C_r$

${ }^n C_x={ }^n C_y \Rightarrow x=y$ or $x+y=n$

If $n$ is even, then the greatest value of ${ }^n C_r$ is ${ }^n C_{n / 2}$

If $n$ is odd, then the greatest value of ${ }^n C_r$ is ${ }^n C_{\frac{n+1}{2}}$

${ }^n C_0+{ }^n C_r+$ .. $+{ }^n C_n=2^n$

${ }^n C_n+{ }^{n+1} C_n+{ }^{n+2} C_n+$ $+{ }^{2 n1} C_n={ }^{2 n} C_{n+1}$

Combinations under restrictions:
PYQ2023P&CQ2, PYQ2023P&CQ13, PYQ2023MatricesQ13

Number of ways of choosing r objects out of n different objects if p particular objects must be excluded the required number of ways $$= ^{np}C_r$$

Number of ways of choosing r objects out of n different objects if p particular objects must be included (p ≤ r). the required number of ways $$= ^{np}C_{rp}$$

The total number of combinations of n different objects taken one or more at a time $$= 2^n  1$$
Combinations of alike objects:
PYQ2023P&CQ5, PYQ2023P&CQ6
$\quad$ If out of (p + q + r + s) objects, p are alike of one kind, q are alike of a second kind, r are alike of the third kind and s are different, then the total number of combinations is $$(p + 1)(q + 1)(r + 1)2^s  1$$
Division into groups:

The number of ways in which (m + n) different objects can be divided into two unequal groups containing m and n objects respectively is $$\frac{(m+n)!}{m! n!}$$

If m = n, the groups are equal and in this case the number of division is $\frac{(2n)!}{n!n!2!} $; as it is possible to interchange the two groups without obtaining a new distribution.

The number of ways in which mn different objects can be divided equally into m groups if order of groups is not important is $$\frac{m n!}{(n!)^m m!}$$

The number of ways in which mn different objects can be divided equally into m groups if the order of groups is important is $$\frac{(m n)!}{(n!)^m }$$