Permutations and Combinations
Formulae:
Factorial Notation:
Let n be positive integer. Then
,factorial n denoted by n!
is defined as n! = n(n-1)(n-2).
. . . . . . .3.2.1
Permutations:
The different arrangements of a
given number of things by
taking some or all at a time,
are called permutations.
e.g.:- All permutations( or
arrangements)made with the letters
a,b,c by taking two at a time
are (ab,ba,ac,ca,bc,cb)
Numbers of permutations:
Number of all permutations of n
things, taken r at a time is
given by nPr =
n(n-1)(n-2). . .. . . (n-r+1)
= n! / (n-r)!
An Important Result:
If there are n objects of which
p1 are alike of one kind;
p2 are alike of another kind ;
p3 are alike of third kind and
so on and pr are alike of rth kind, such that
(p1+p2+. . . . . . . . pr) = n
Then, number of permutations of
these n objects is:
n! / (p1!).(p2!). . . . .(pr!)
Combinations:
Each of different groups or
selections which can be formed by
taking some or all of a number
of objects, is called a combination.
e.g.:- Suppose we want to select two out of
three boys A,B,C .
then ,possible selection are AB,BC
& CA.
Note that AB and BA represent the same
selection.
Number of Combination:
The number of all combination
of n things taken r at a time is:
nCr = n! / (r!)(n-r)!
= n(n-1)(n-2). . . . . . . tor factors
/ r!
Note: nCn = 1 and nC0 =1
An Important Result:
nCr = nC(n-r)
To exercise problems regarding permutations & combinations visit here
Probability
Introduction:
Experiment:
An operation which can produce
some well-defined outcome is called an experiment.
Random Experiment:
An experiment in which all
possible out comes are known and the exact output cannot be predicted in
advance is called a random experiment. EX:
1) Rolling an unbiased dice.
2) Tossing a fair coin.
3) Drawing a card from a pack
of well-shuffled cards .
4)Picking up a ball of certain color
from a bag containing balls of different colors.
Details:
1) When we thrown a coin ,then
either a Head(H) or a Tail(T)appears.
2)A dice is a solid cube
,having 6 faces, marked 1,2,3,4,5,6respectively. When we throw a die ,the
outcome is the number that appears on its upper face.
3)A pack of cards has 52 cards.
It has 13 cards of each suit, namely
spades, clubs, hearts and diamonds.
Cards of spades and clubs are black cards.
Cards of hearts and diamonds
are red cards.
There are four honors of each
suit.
These are Aces, Kings, queens
and Jacks.
These are called Face cards.
Sample Space:
When we perform an experiment
,then the set of S of all possible outcomes is called the Sample space .
Event: Any subset of a sample space is
called an Event.
Probability of occurrence of an
Event:
Let S be the sample space.
Let E be the Event.
Then E c S i.e. E is subset of
S then
probability of E p(E) =n(E)/n(S).
Results on Probability:
1)P(S) =1.
2)0 < P(E) < 1
probability of an event lies
between 0 and 1.
Max value of probability of an
event is one.
3)For any events A and B we
have .
P(AUB) =P(A) +P(B) -P(A n B ).
4)If A denotes (not -A) then
P (A) =1-P(A)
P(A)+P(A) =1.
To Exercise problems in probability visit here
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