1. PERCENTAGE

Out of 100 equal parts, each part is known as its hundredth part.
Thus 6 hundredths = 6/100, 17 hundredths = 17/100 etc.
By a certain percentage, we mean that many hundredths.
We denote x percent by x% .
Thus x% = x hundredths = x/100
54% = 54 hundredths = 54/100
We have x% = x/100
Rule: For converting a percentage into a fraction, divide if by 100 and remove the sign %
Rule: A percentage can be expressed as ratio with first term equal to given percentage and second term 100.
Rule: First convert the given percentage as a fraction and then convert this fraction into decimal form.
Rule: First write the ratio as a fraction and then multiply with 100 to get percentage.
Rule: Convert the given decimal into a fraction and then multiply by 100 to get percentage.
COST PRICE : The price at which an article is purchased is called cost price (C.P.).
SELLING PRICE : The price at which an article is sold is called selling price (S.P.).
PROFIT : If the selling price of an article is more than the cost price then the seller is said to have made a profit, where Profit = S.P. – C.P.
LOSS : If the selling price of an article is less than the cost price then the seller is said to have incurred a loss, where Loss = C.P. – S.P.
PERCENTAGE PROFIT : It is always reckoned on C.P.
PERCENTAGE LOSS : It is always reckoned on C.P
.
COST PRICE IN TERMS OF PROFIT :
COST PRICE IN TERMS OF LOSS :
SELLING PRICE IN TERMS OF PROFIT :
SELLING PRICE IN TERMS OF LOSS :
SALE at equal gain and loss % : When a person sells two identical items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by:
RATIO : Ratio of two quantities a and b is given by a:b. Here a and b are called antecedent and consequent respectively.
PROPORTION : When two ratio are equal to each other then they are said to be proportional.
Here a and d are called extreme terms while b and c are called mean terms.
Numerically, Product of Means = Product of Extremes
Compounded Ratio : The compounded ratio of a:b, c:d and e:f is ace:bdf.
if 
ALLIGATION: It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.
If an operation can be performed in ‘m’ different ways; following which a second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in m × n ways. This can be extended to any finite number of operations.
If an operation can be performed in ‘m’ different ways and another operation, which is independent of the first operation, can be performed in ‘n’ different ways. Then either of the two operations can be performed in (m + n) ways. This can be extended to any finite number of mutually exlusive operations.
If n is a natural number then the product of all natural numbers upto n is called factorial n and it is denoted by
n ! or .
Thus n ! = n (n – 1) (n – 2) ...... 3.2.1.
It is obvious to note that
= n
= n (n – 1)
= n (n – 1) (n – 2) etc.
Note : 0 ! = 1, 1 ! = 1, 2 ! = 2, 3 ! = 6, 4 ! = 2, 5 ! = 120, 6 ! = 720
Each of the different arrangements which can be made by taking some or all of a number of things is called a permutation.
(i) The number of permutations of n different things, taking r at a time is denoted by ^{n}P_{r} or P(n, r)
then (0 < r < n)
= n(n – 1) (n – 2) ..... (n – r + 1), n Ïµ N and r Ïµ W
(ii) The number of arrangements of n different objects taken all at a time is ^{n}P_{n} = n !
Note : ^{n}P_{1} = n, ^{ n}P_{r} = n. ^{n–1}P_{r–1}, ^{n}P_{r} = (n–r+1). ^{n}P_{r–1}, ^{n}P_{n} = ^{n}P_{n–1}
(i) The number of permutations of n things taken all at a time, p are alike of one kind, q are alike of second kind and r are alike of a third kind and the rest n – (p + q + r) are all different is
(ii) The number of permutations of n different things taken r at a time when each thing may be repeated any number of times is nr.
=^{ n–r}P_{n–r} =
r . ^{n – 1}P_{r – 1}
n – 1Pr
m ! × (n – m + 1)!
n ! – m ! × (n – m + 1) !
(i) Arrangement round a circular table: The number of circular permutations of n different things taken all at a time is (n – 1) !, if clockwise and anticlockwise orders are taken as different.
(ii) Arrangement of beads or flowers (all different) around a circular necklace or garland: The number of circular permutations of n different things taken all at a time is (n –1) !, if clockwise and anticlockwise orders are taken as not different.
(iii) Number of circular permutations of n different things taken r at a time:
(iv) Restricted Circular Permutations: When there is a restriction in a circular permutation then first of all we shall perform the restricted part of the operation and then perform the remaining part treating it similar to a linear permutation.
Each of the different groups or selections which can be made by some or all of a number of given things without reference to the order of the things in each group is called a combination.
Problems of permutations  Problems of combinations 
1. Arrangements  Selections, choose 
2. Standing in a line, seated in a row  Distributed group is formed 
3. Problems on digits  Committee 
4. Problems on letters from a word  Geometrical problems 
7.1 SELECTION OF OBJECTS WITHOUT REPETITION
The number of combinations of n different things taken r at a time is denoted by ^{n}C_{r }or C (n, r) or
Then ; (0 < r < n)
; n Ïµ N and r Ïµ W
If r > n, then ^{n}C_{r }= 0.
Some important results :
(i) ^{ n}C_{n} = 1, ^{n}C_{0} = 1,  (ii) ^{n}C_{r} = 
(iii) ^{n}C^{r} =^{ n}C_{n–r},  iv) ^{n}C_{x} = ^{n}C_{y } => x + y = n, 
(v) nC_{r} + ^{n}C_{r–1} = ^{n+1}C_{r}  (vii) ^{n}C_{r }= (n – r + 1) ^{n}C_{r–1}, 
(vi) ^{n}C_{r} = . ^{n–1}C_{r–1},  
(viii) ^{n}C_{1} = ^{n}C_{n–1} = n 
The total number of selections of r things from n differents things when each thing may be repeated any number of times is
^{n + r–1}C_{r}
(i) The number of combinations of n different things taken r at a time,
(ii) The number of arrangements of n distinct objects taken r at a time so that k particular object are
(iii) The number of combinations of n objects, of which p are identical, taken r at a time is
The number of ways (or combinations) of n different things selecting at least one of them is
^{n}C_{1} + ^{n}C_{2} + ^{n}C_{3} +......+ ^{n}C_{n} = 2^{n} – 1. This can also be stated as the total number of combination of n different things.
= (p + 1) (q + 1) (r + 1) – 1
If out of (p + q + r + t) things, p are alike one kind, q are alike of second kind, r are alike of third kind and t are different, then the total number of combinations is
(p + 1) (q + 1) (r + 1) 2^{t} – 1
7.7 Divisors of a given natural number :
Let , where p_{1}, p_{2}, p_{3}, .... pk are different primes and α_{1}, α_{2}, α_{3}, ........, α_{k} are natural numbers then :
= (α_{1} + 1) (α_{2} + 1) (α_{3} + 1) .... (α_{k} + 1)
= (α_{1} + 1) (α_{2} + 1) (α_{3} + 1) .... (α_{k} + 1) – 2
= (α_{1} + 1) (α_{2} + 1) (α_{3} + 1) .... (α_{k} + 1) – 1
= (p_{1}^{0} + p_{1}^{1} + p_{1}^{1} + ....+ ) (p_{2}^{0} + p_{2}^{1} + p_{2}^{2} + ....+ )...... (p_{k}^{0} + p_{k}^{1} + p_{k}^{2} + ....+ )
(α_{1} + 1) (α_{2 }+ 1) .... (α_{k} + 1), If N is not a perfect square
[(α_{1} + 1) (α_{2} + 1) .... (α_{k} + 1) + 1], If N is a perfect square
Any change in the given order of the thing is called a Derangement.
(i) If n items are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is
(ii) If n things are arranged at n places then the number of ways to rearrange exactly r things at right places is
The following table shows the conditions of divisibility of some numbers
Divisible by  Condition 
2  whose last digit is even 
3  sum of whose digits is divisible by 3 
4  whose last two digits number is divisible by 4 
5  whose last digit is either 0 or 5 
6  which is divisible by both 2 and 3 
8  whose last three digits number is divisible by 8 
9  sum of whose digits is divisible by 9 
25  whose last two digits are divisible by 25 
(i) For given n different digits a_{1}, a_{2}, a_{3} ......., a_{n} the sum of the digits in the unit place of all numbers fromed (if numbers are not repeated) is
(a_{1} + a_{2} + a_{3} + ...... + a_{n}) (n – 1)!
i.e. (sum of the digits) (n – 1) !
(ii) Sum of the total numbers which can be formed with given n different digits a_{1}, a_{2}, a_{3} ....... a_{n} is
(a_{1} + a_{2} + a_{3} + ..... + a_{n}) (n – 1) ! . (111 ......... n times)
If there are n points in a plane of which m(< n) are collinear, then
1 + Σn
Ordered Pair: Two elements a and b listed in a specific order , form an Ordered Pair ,denoted by (a , b).
In an ordered pair (a , b), we call a as the first component and b as the second component.
By changing the position of the components , the ordered pair is changed, i.e. (1, 4) (4, 1).
Also, (a , b) = (c , d) => a = c and b = d
Thus, (x , y) = (2 , 7) => x = 2 and y = 7
Cartesian Product of two sets : Let A and B be two nonempty sets. Then, their Cartesian product A × B is the set of all ordered pairs (a , b) such that a A and b B.
i.e. A × B = {(a, b) : a A and b B}
Relation: Let A and B be two nonempty sets. Then, every subset of A × B is called a relation from A to B.
i.e. If R ⊂ A × B, then R is relation from A to B.
Also, if (a , b) Ïµ R, we say that a is related to b and we write aRb.
Representation of a Relation :
(i) Roster Form: A relation represented by the set of all ordered pairs contained in it, is said to be in roster form.
Example: If A = {1, 9, 16, 25 } and B = {1, 2, 3, 4, 5}, then a relation R from A to B defined as 'is the square' of can be represented in the roster form as :
R = {(1, 1), (9, 3), (16, 4), (25, 5)}.
(ii) Arrow Diagram : Let R be a relation from A to B. We draw closed bounded figures to represent the sets A and B and write their elements in the corresponding figures. Then, we draw arrows from A to B to indicated the pairing of the corresponding elements related to each other. Thus, we can show the relation given in example 2 by the arrow diagram as shown below.
(iii) SetBuilder Form : A relation R from A to B is said to be in a SetBuilder Form, when written as:
R = {(a, b) : a Ïµ A, b Ïµ B and a is connected with b by a given rule}
The relation given in example 2 can be represented in the setbuilder form as :
R = {(a, b) : a Ïµ A, b Ïµ B and a = b2}
Domain and Range of a Relation : Let R be a relation from A to B. Then
Domain (R) = Set of first component of all ordered pairs in R.
Range (R) = Set of second component of all ordered pairs in R.
Range of Relation: The property that tells us how the first component is related to the second component of each ordered pair in R, is called the rule of the relation.
Function or Mapping : Let A and B be two nonempty sets. Then, a function or a mapping f from A to B is a rule which associates to each element x Ïµ A, a unique element f(x) Ïµ B, called the image of x. If f is a function from A to B, then we write f : A → B.
Remarks: For f to be a function from A to B :
(i) every element in A must have its image in B.
(ii) no element in A must have more than one images.
Representation of a function:
(i) Arrow Diagram : Let f be a relation from A to B. We draw closed bounded figures to represent the sets A and B and write their elements in the corresponding figures. Then, we draw arrows from A to B joining each elements of A with its
corresponding image in B. The function given in the above example can be represented by the arrow diagram as shown above.
(ii) Roster Form : Let f be a function from A to B. We form ordered pairs of all the elements of A with their corresponding images in B. Then, the function f is represented as the set of all such ordered pairs The function given in the above example can be represented in the roster form as f = {(1, 2), (2, 4), (3, 6)}.
(iii) Equation Form : Let f be a function from A to B. If f be represented in terms of the rule of association , then it is said to be in the equation form.
In the above example, the rule of association is given by f (x) = 2x . Let y Ïµ B be the image of an element x Ïµ A.
Then, We have y = 2x
Then, y = 2x is the representation of function f in the equation form.
Domain, CoDomain and Range of A Function : Let f be a function from A to B . Then, we define:
(a) Domain (f) = A (b) Codomain (f) = B
(c) Range (f) = Set of all images of elements of A
Function as a Relation: Let A and B be two nonempty sets and R be a relation from A to B. Then R is called a function from A to B , if domain (R) = A and no two ordered pairs in R have the same first components.
Remark: Every function is a relation. But, every relation need not be a function.
REAL VALUED FUNCTIONS: A rule f which associates to each real number x, a unique real number f(x), is called a real valued function. Here, f(x) is an expression in x.
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