FUNDAMENTAL PRINCIPLE OF COUNTING
If an operation can be performed in 'm' different ways and another
operation in 'n' different ways then these two operations can be
performed one after the other in 'mn' waysIf an operation can be
performed in 'm' different ways and another operation in 'n'
different ways then either ofthese two operations can be performed
in 'm+n' ways.(provided only one has to be done) 

This principle can be extended to any number of
operations
FACTORIAL
'n'
The continuous product of the first 'n' natural numbers is
called factorial n and is deonoted by n! i.e, n! = 1×2×3x …..
x(n1)xn.
PERMUTATION
An
arrangementthat can be formed by taking some or all
of a finite set of things (or objects) is called a
Permutation.Order of the things is very important
in case of permutation.A permutation is said to be a
Linear
Permutation if the objects are arranged in a line. A
linear permutation is simply called as a permutation.A permutation
is said to be a
Circular Permutation if the
objects are arranged in the form of a circle.The number of (linear)
permutations that can be formed by taking r things at a time from a
set of n distinct things
is denoted by
.%
NUMBER OF PERMUTATIONS UNDER
CERTAIN CONDITIONS
1. Number of permutations of n different things, taken r at a
time, when a particular thng is to be always included in each
arrangement , is .
2. Number of permutations of n different things, taken r at a
time, when a particular thing is never taken in each arrangement is
.
3. Number of permutations of n different things, taken all at a
time, when m specified things always come together is .
4. Number of permutations of n different things, taken all at a
time, when m specified never come together is .
5. The number of permutations of n dissimilar things taken r at
a time when k(< r) particular things always occur is .
6. The number of permutations of n dissimilar things taken r at
a time when k particular things never occur is .
7. The number of permutations of n dissimilar things taken r at
a time when repetition of things is allowed any number of times is
8. The number of permutations of n different things, taken not
more than r at a time, when each thing may occur any number of
times is .
9. The number of permutations of n different things taken not
more than r at a time .
+ PERMUTATIONS OF
SIMILAR THINGS+ The number of
permutations of n things taken all tat a time when p of them are
all alike and the rest are all different is .If p things are alike of one type, q things
are alike of other type, r things are alike of another type, then
the number of permutations with p+q+r things is . 

CIRCULAR PERMUTATIONS
}1. The number of circular permutations of n dissimilar things
taken r at a time is .
2. The number of circular permutations of n dissimilar things
taken all at a time is .
3. The number of circular permutations of n things taken r at a
time in one direction is .
4. The number of circular permutations of n dissimilar things in
clockwise direction = Number of permutations in anticlockwise
direction = .
COMBINATION
A selection that can be formed by taking some or all
of a finite set of things( or objects) is called a
Combination
The number of combinations of n dissimilar things taken r at a
time is denoted by .
1.
2.
3.
4.
5. The number of combinations of n things taken r at a time in
which
a)s particular things will always occur is .
b)s particular things will never occur is .
c)s particular things always occurs and p particular things
never occur is .
DISTRIBUTION OF
THINGS INTO
GROUPS
1.Number of ways in which (m+n) items can be divided into two
unequal groups containing m and n items is .
2.The number of ways in which mn different items can be divided
equally into m groups, each containing n objects and the order of
the groups is not important is
3.The number of ways in which mn different items can be divided
equally into m groups, each containing n objects and the order of
the groups is important is .
4.The number of ways in which (m+n+p) things can be divided into
three different groups of m,n, an p things respectively is
5.The required number of ways of dividing 3n things into three
groups of n each =.When the order of
groups has importance then the required number of ways=
DIVISION OF IDENTICAL OBJECTS
INTO GROUPS
The total number of ways of dividing n identical items among r
persons, each one of whom, can receive 0,1,2 or more items
is
}The number of nonnegative integral solutions of the equation
.
The total number of ways of dividing n identical items among r
persons, each one of whom receives at least one item is
The number of positive integral solutions of the equation
.
The number of ways of choosing r objects from p objects of one
kind, q objects of second kind, and so on is the coefficient of
in
the expansion
he number of ways of choosing r objects from p objects of one
kind, q objects of second kind, and so on, such that one object of
each kind may be included is the coefficient of is the coefficient
of in the expansion
.
SUM OF THE NUMBERS
Sum of the numbers formed by taking all the given n digits
(excluding 0) is
Sum of the numbers formed by taking all the given n digits
(including 0) is
Sum of all the rdigit numbers formed by taking the given n
digits(excluding 0) is
%
%{fontfamily:verdana}Sum of all the rdigit numbers
formed by taking the given n digits(including 0) is
DEARRANGEMENT:
The number of ways in which exactly r letters can be placed in
wrongly addressed envelopes when n letters are placed in n
addressed envelopes is
.
The number of ways in which n different letters can be placed in
their n addressed envelopes so that al the letters are in the wrong
envelopes is
.
IMPORTANT RESULTS TO REMEBER
In a plane if there are n points of which no three are
collinear, then
1. The number of straight lines that can be formed by joining
them is .
2. The number of triangles that can be formed by joining them is
.
3. The number of polygons with k sides that can be formed by
joining them is .
In a plane if there are n points out of which m points are
collinear, then
1. The number of straight lines that can be formed by joining
them is .
2. The number of triangles that can be formed by joining them is
.
3. The number of polygons with k sides that can be formed by
joining them is .
Number of rectangles of any size in a square of n x n is
In a rectangle of p x q (p < q) number of rectangles of any
size is
In a rectangle of p x q (p < q) number of squares of any size
is
n straight lines are drawn in the plane such that no two lines
are parallel and no three lines three lines are concurrent. Then
the number of parts into which these lines divide the plane is
equal to .
Image Credits: cristic,
cosmolallie,
farouqtaj,
churl
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