Addition law of counting :

If a work $W$ consists of two parts $W_1,W_2$ of which one part can be done in $m$ ways and the other part in $n$ ways thern , the work $W$ can be done in $m+n$ ways, if by doing any of the parts the work $W$ is done.

Multiplication law of counting :

If a work $W$ consists of two parts $W_1,W_2$ of which one part can be done in $m$ ways and the other part in $n$ ways thern the work $W$ can be done in $mn$ ways, if both the parts has to be done one after the other to do the work $W$.

Note :

If a work is to be done under some restriction then

the number of ways to do the work under the restriction = (the number of ways to do the work without restriction) - (the number of ways to do the work under opposite restriction).

The number of permutations (arrangements) of $n$ different things taking $r$ at a time

$={}^n{P}_r=\frac{n!}{(n-r)!}\text{ where }n!=\mathrm{1.2.3}\dots n\text{. }$

The number of permutations of $n$ things taking all at a time of which $p$ things are identical, $q$ things are identical of another type and the rest are different

$=\frac{n!}{p!q!}$

Circular Permutation :

The number of arrangements of $n$ diffirent things around a closed curve

$=(n-1)!$ if clockwise and anticlockwise arrangements are considered different,

= $\frac{1}{2}(n-1)!$ if clockwise and anticlockwise arrangements are considered identical.

1. The number of combinations (selections) of $n$ different things taking $r$ at a time

$$ ={ }^n C_r=\frac{n !}{r !(n-r) !} \text {. } $$

2. The total number of selections of one or more objects from $n$ different objects

$$ =2^n-1={ }^n C_1+{ }^n C_2+{ }^n C_3+\ldots+{ }^n C_n $$

3. The total number of selections of any number of things from $n$ identical things

$= n+1$ (when selection of 0 things is allowed)

= $n$ (when at least one thing is to be selected).

4. The total number of selections from $p$ like things, $q$ like things of another type and $r$ distinct things

$\begin{aligned}=(p+1)(q+1) 2^r-1 & \text { (if at least one thing is } \\ & \text { to be selected) } \\ = (p+1)(q+1) 2^r-2 & \text { (if none or all cannot be } \\ & \text { selected) }\end{aligned}$

5. The total number of selections of $r$ things from $n$ different things when each thing can be repeated unlimited number of times $={ }^{n+r-1} C_r$

1. The number of ways to distribute $n$ diffirent things between two persons, one receiving $p$ things and the other $q$ things, where $p+q=n$

$$ \begin{aligned} & ={ }^n C_p \times{ }^{n-p} C_q \\\\ & =\frac{n !}{p !(n-p) !} \times \frac{(n-p) !}{q !(n-p-q) !} \end{aligned} $$

$$ =\frac{n !}{p ! q !} \left\{ \because n=p+q \right\} $$

Similarly for 3 persons, the number of ways

$$ =\frac{n !}{p ! q ! r !} \text {, where } p+q+r=n \text {. } $$

2. The number of ways to distribute $m \times n$ different things among $n$ persons equally $=\frac{(m n) !}{(m !)^n}$.

3. The number of ways to divide $n$ different

things into three bundles of $p, q$ and $r$ things $=\frac{n !}{p ! q ! r !} \cdot \frac{1}{3 !}$.

4. The number of ways to divide $m \times n$ different

things into $n$ equal bundles $=\frac{(m n) !}{(m !)^n} \cdot \frac{1}{n !}$.

5. The total number of ways to divide $n$ identical things among $r$ persons

$$ ={ }^{n+r-1} C_{r-1} $$

6. The total number of ways to divide $n$ identical things among $r$ persons so that each gets at least one

$$ ={ }^{n-1} C_{r-1} . $$

A Useful Notation :

$n !=n(n-1)(n-2)$...... 3. 2. 1;

$n !=n .(n-1) !$ where $n \in W$

$0 !=1 !=1$

$(2 \mathrm{n}) !=2^{\mathrm{n}} \cdot \mathrm{n} ![1.3 \cdot 5 \cdot 7 \ldots \ldots . .(2 \mathrm{n}-1)]$

Note that :

(i) Factorial of negative integers is not defined.

(ii) Let $\mathrm{p}$ be a prime number and $\mathrm{n}$ be a positive integer, then exponent of $p$ in $n$! is denoted by $E_p(n$!) and is given by

$$ E_p(n !)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\left[\frac{n}{p^3}\right]+\ldots . $$

Let $\mathrm{N}=\mathrm{p}^{\mathrm{a}} \cdot \mathrm{q}^{\mathrm{b}} \cdot \mathrm{r}^{\mathrm{c}} \ldots \ldots$ where $\mathrm{p}, \mathrm{q}, \mathrm{r} \ldots \ldots \ldots$ are distinct primes $ \mathrm{a}$, $\mathrm{b}, \mathrm{c} \ldots \ldots$ are natural numbers then :

(a) The total numbers of divisors of $N$ including 1 and N is

$=(a+1)(b+1)(c+1) \ldots \ldots$

(b) The sum of these divisors is

$\left(p^0+p^1+p^2+\ldots+p^a\right)$ $\left(q^0+q^1+q^2+\ldots .+q^b\right)\left(r^0+r^1+r^2+\ldots .+r^c\right) \ldots$

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

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

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

(d) 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 $2^{\mathrm{n}-1}$ where $\mathrm{n}$ is the number of different prime factors in N.

Number of ways in which $\mathrm{n}$ letters can be placed in $\mathrm{n}$ directed envelopes so that no letter goes into its own envelope is

$$ =n !\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\ldots \ldots . .+(-1)^n \frac{1}{n !}\right] $$

(a) Number of rectangles of any size in a square of size $n \times n$

is $$\sum\limits_{r = 1}^n {{r^3}} $$ number of squares of any size is $$\sum\limits_{r = 1}^n {{r^2}} $$.

(b) Number of rectangles of any size in a rectangle of size $\mathrm{n} \times \mathrm{p}$ $(n < p)$ is $\frac{n p}{4}(n+1)(p+1)$

number of squares of any size is $\sum\limits_{r=1}^n(n + 1 - r)(p + 1 - r)$

(c) If there are $\mathrm{n}$ points in a plane of which $\mathrm{m}(<\mathrm{n})$ are collinear :

(i) Total number of lines obtained by joining these points is ${ }^{\mathrm{n}} \mathrm{C}_2-{ }^{\mathrm{m}} \mathrm{C}_2+1$

(ii) Total number of different triangle ${ }^n C_3-{ }^m C_3$

(d) Maximum number of point of intersection of $n$ circles is ${ }^{\mathrm{n}} \mathrm{P}_2$ and n lines is ${ }^{\mathrm{n}} \mathrm{C}_2$.