**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)

**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.

**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 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 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

.

**TOTAL NUMBER OF COMBINATIONS**

1.The total number of combinations of things taken any number at a time when things are alike of one kind, things are alike of second kind….things are alike of kind, is .

2.The total number of combinations of

things taken one or more at a time when

things are alike of one kind,

things are alike of second kind….

things are alike of

kind, is

.

**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 .