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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' ways. If 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) |
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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!=1×2x3x…..x(n-1)xn.
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PERMUTATION An arrangement that 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 |
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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 .
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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 |
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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 clock-wise direction = Number of
permutations in anticlock-wise 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 non-negative 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
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TOTAL NUMBER OF COMBINATIONS
1.The total number of combinations of
2.The total number of combinations of
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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 r-digit numbers formed by taking
the given n digits(excluding 0) is
Sum of all the r-digit numbers formed by taking
the given n digits(including 0) is
DE-ARRANGEMENT:
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
Number of squares 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 .
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