1. If all given objects are different and repetition is allowed then the number of arrangements.

*n*

^{r}2. If all given objects are different, but repetition is not allowed then the number of arrangements.

3. If there are kinds of duplicate elements, then the number of arrangements by using all elements in each arrangement.

4. Circular arrangements with difference in clock-wise and anticlockwise is given by

*n*-1)!

5. If there is no regards of order, it is called combination, denoted by

_{n}C

_{r}, C(

*n;r*) or C

_{r}

^{n}•

*Fundamental principle of counting*If an event can occur in

*m*different ways, following which another event can occur in

*n*different ways, then the total number of occurrence of the events in the given order is

*m×n*.

•The number of permutations of

*n*different things taken

*r*at a time, where repetition is not allowed, is denoted by

^{n}P

_{r}, and is given by

where 0≤

*r*≤

*n*.

•

*n*!=1×2×3×…×

*n*

•

*n*!=

*n*×(

*n*-1)!

•The number of permutations of

*n*different things, taken

*r*at a time, where repeatition is allowed, is

*n*.

^{r}•The number of permutations of

*n*objects taken all at a time, where

*p*

_{1}objects are of first kind,

*p*

_{2}objects are of the second kind, …,

*p*

_{k}objects are of the

*k*-th kind and rest, if any, are all different is

•The number of combinations of

*n*different things taken

*r*at a time, denoted by

^{n}C

_{r}, is given by

In this Chapter, we studied about the axiomatic approach of probability. The main features of this Chapter are as follows:

•Sample space: The set of all possible outcomes

•Sample points: Elements of sample space

•Event: A subset of the sample space

•Impossible event: The empty set

•Sure event: The whole sample space

•Complementary event or ‘not event’ : The set

*A*or

*S-A*

•Event

*A*or

*B*: The set

*A∪B*

•Event

*A*and

*B*: The set

*A∩B*

•Event

*A*and not

*B*: The set

*A-B*

•Mutually exclusive event:

*A*and

*B*are mutually exclusive if

*A∩B*=Ø.

•Exhaustive and mutually exclusive events: Events

*E*

_{1},

*E*

_{2}, …,

*E*

_{n}are mutually exclusive and exhaustive if

*E*

_{1}∪

*E*

_{2}∪…∪

*E*

_{n}=

*S*and

*E*

_{1}∩

*E*

_{2}∩…∩

*E*

_{n}=Ø.

•Probability: The value of a probability P must be in 0≤

*P*≤1.

• Equally likely outcomes: All outcomes with equal probability

• Probability of an event: For a finite sample space with equally likely outcomes. Probability of an event

*A*)=n(

*A*)/n(

*S*)

where n(

*A*)=number of elements in the set

*A*, n(

*S*)=number of elements in the set

*S*.

•If

*A*and

*B*are any two events, then

*A*or

*B*)=P(

*A*)+P(

*B*)-P(

*A*and

*B*)

equivalently,

*A∪B*)=P(

*A*)+P(

*B*)-P(

*A∩B*)

•If

*A*and

*B*are mutually exclusive, then P(

*A*or

*B*)=P(

*A*)+P(

*B*)

•If

*A*is any event, then P(

*n*ot

*A*)=1-P(

*A*)

The basic counting principle:

The number of ways of making several decisions in succession (call them

*m*

_{1},

*m*

_{2}and

*m*

_{3}etc…) is determined by multiplying the numbers of choices that can be made in each decision.

*m*

_{1}×

*m*

_{2}×

*m*

_{3}×…

Permutations

● The number of permutations of *m* different items is *m*!

● The number of permutations of *m* items of which:

*a* are alike, *b* are alike, *c* are alike is:

● The number of permutations of in items taken

*n*at a time, when each of the items may be repeated any number of times, is:

*m×m×m×m*× … to

*n*factors =

*m*times.

^{n}● The number of ways that

*m*items taken

*n*at a time can be arranged, is

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