Permutations and Combinations Formulas

Permutations and Combinations:

Factorial Notation:

Let n be a positive integer.
Then, factorial n, denoted n! is defined as:
n! = n(n - 1)(n - 2) ... 3.2.1.

Permutations:

Each of the arrangement which can be made by taking some are all of a number of things is called a permutation
permutation means "arrangement" i.e, the order of the "objects" is important.
The permutation of three things a,b and c taken two at a time are ab, ba, ac, ca, cb, bc .Since the order in which the object are taken is important
ab and ba are counted as a two different permutation.
The word "Permutation" and "arrangement" are synonymous and can be used interchangeably
.The different arrangements of a given number of things by taking some or all at a time, are called  permutations.

Examples:

All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)

Number of Permutations:

The number of permutations of  n things taken r at a time is denoted by:

Combinations:

Each of the different groups or selections which can be made by taking some or all of a number of objects is called a combination.
In combination,the order in which the object are taken is not considered
The combination of three objects-a,b,and c taken two at a times are ab, bc, ca. here ab, ba are not considered separately because the order in which a and b are taken is not important

Number of Combinations:

The number of all combinations of n things, taken r at a time is:

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Volume and Surface Area formulas

Rectangle

Perimeter:
P = L+ L+ w + w
or
P = 2(l + w)
Area:
A = lw

Parallelogram:

Perimeter:

p=b+b+c+c

        or

P=2b+2c

Area:

A=bh

Triangle:

Perimeter:

P = q + w +e

Area:
A=bh / 2
    or    
A=1/2*b*h

Trapezoid:

Perimeter:
p=a + b + c + d

Area:
A=(a + b)h/2
       or
A=1/2(a + b)h

Circle:

Cylinder:

 Sphere:

Cone:

Rectangular prism:

Triangular prism:

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Logical Deduction

Logical Deduction:

The phenomenon of deriving a conclusion from a single proposition or a set of given propositions, is known as logical deduction.
The given propositions are also referred to as the premises.
.

Immediate Deductive Inference :

Here, conclusion is deduced from one of the given propositions, by the three ways
1)conversion,
2)obversion and
3)contraposition.

1. Conversion: The Conversion proceeds with interchanging the
subject term and the predicate term i.e.
the subject term of the premise becomes the predicate term of the
conclusion and the predicate term of the premise becomes the subject of the conclusion.
The given proposition is called convertend, whereas the conclusion drawn from it is called its converse.

2. Obversion: In obversion, we change the quality of the proposition and replace the predicate term by its complement.

3. Contraposition: To obtain the contrapositive of a statement, we first replace the subject and predicate terms in the proposition and then exchange both these terms with their complements.


Mediate Deductive Inference (SYLLOGISM): First introduced by Aristotle, a Syllogism is a deductive argument in which conclusion has to be drawn from two propositions referred to as the premises.

Example:
1. All lotus are flowers.

2. All flowers are beautiful.

3. All lotus are beautiful.

Clearly, the propositions 1 and 2 are the premises and the proposition 3, which follows from the first two propositions, is called the conclusion.


Term : In Logic, a term is a word or a combination of words, which by itself can be used as a subject or predicate of a proposition.

Syllogism is concerned with three terms :

1. Major Term : It is the predicate of the conclusion and is denoted by P (first letter of 'Predicate').

2. Minor Term: It is the subject of the conclusion and is denoted by S (first letter of 'Subject').

3. Middle Term: It is the term common to both the premises and is denoted by M (first letter of 'Middle').

Example:
Premises:
1. All dogs are animals.

2. All tigers are dogs.

Conclusion :
All tigers are animals.
Here 'animals' is the predicate of the conclusion and so,.it is the major term. P.
'Tigers' is the subject of the conclusion and so, it is the minor term, S.

'Dogs' is the term common to both the premises and so, it is the middle term, M.

Major And Minor Premises : Of the two premises, the major premise is that in which the middle term is the subject and the minor premise is that in which the middle term is the predicate.

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Ratio and Proportion

Ratio and Proportion

1. What is Ratio:

The ratio of two quantities a and b in the same units, if the fraction
ab
and we write it as a : b.

In the ratio a : b, we call a as the first term which is also know as antecedent and b, the second term which is also called consequent.

2. What is Proprotion:

Equality of two ratios is called proportion.

The equality of two ratios is called proportion.

If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.

Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.
Thus, a : b :: c : d <=> (b x c) = (a x d).

3. Fourth, third and mean proportional

i). Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

ii). Third Proportional:

a : b = c : d, then c is called the third proportion to a and b.

iii). Mean Proportional:

Mean proportional between a and b is
√ab

4. Comparison of Ratios and Compounded Ratio

i). Comparison of Ratios:

When we say that a:b > c:d, then it means
a/b>c/d

ii). Compounded Ratio:

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf)

Please note it is ratio of first ratio term of every ratio and second ratio term of every ratio

5. Important results of Ratio:

i). Duplicate ratio of (a:b) is
(a2:b2)


ii). Sub-duplicate ratio of (a : b) is
(√a:√b)


iii). Triplicate ratio of (a : b) is
      (a3:b3)
      a3 is (a power of 3)
      b3 is (b power of 3)


iv). Sub Triplicate ratio of (a : b) is
     (a1/3:b1/3)
     a1/3 is (a power of 1/ 3)
     b1/3 is (b power of 1/3)

v).
if a/b=c/d
then a+b/a−b = c+d/c−d .,

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Number and Ranking

Number and Ranking

1)formula:

Total Number(Items) of arranged are:
n=T(L)+B(R)-1
or
n=L+R-1

T=Top
B=Bottom
L=Left
R=Right

2)formula:

Rank from the Top
T(L)=(n+1)-B(R)
Rank from the Bottom
B(R)=(n+1)-T(L)

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Number Series

Number Series

Some Basic Formula:

(a + b)(a - b) = (a2 - b2)

(a + b)2 = (a2 + b2 + 2ab)

(a - b)2 = (a2 + b2 - 2ab)

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

(a3 + b3) = (a + b)(a2 - ab + b2)

(a3 - b3) = (a - b)(a2 + ab + b2)

(a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)

When a + b + c = 0, then a3 + b3 + c3 = 3abc.

The nth Term

The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next.

Constant Difference Sequences:

nth term = dn + (a - d)
Where d is the difference between the terms, a is the first term and n is the term number.

Changing Difference Sequences:

nth term = a + (n - 1)d + ½(n - 1)(n - 2)c
Where d is the difference between the terms, a is the first term and n is the term number
This time there is a letter c which stands for the second difference (or the difference between the differences and d is just the difference between the first two numbers.

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Blood Relations

Blood Relations

Mother's or father's son     -   Brother
Mother's or father's daughter -Sister
Mother's or father's brother -Uncle
Mother's or father's sister     - Aunt
Mother's or father's father         Grandfather
Mother's or father's mother- Grandmother
Son's wife                              -   Daughter-in-law
Daughter's husband               -  Son-in-law
Husband's or wife's sister       -  Sister-in-law
Husband's or wife's brother- Brother-in-law
Brother's son                        - Nephew
Brother's daughter                - Niece
Uncle or aunt's son or daughter- Cousin
Sister's husband                - Brother-in-law
Brother's wife                        - Sister-in-law
Grandson's or Grand daughter's daughterGreat Grand daughter

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Area formulas and concepts

1. Area formulas and concepts

I. Results on Triangles:
1. Sum of the angles of a triangle is 180.
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem:

In a right-angled triangle 

(Hypotenuse)2=(Base)2+(Perpendicular)2

Please Note, this is very important theorem which not only helps solving the questions in area but also in other chapters like heights and distances etc.

4. The line joining the mid-point of a side of a triangle to the positive vertex is called the median.

5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the medians in the ratio 2 : 1.

6. In an isosceles triangle, the altitude from the vertex bisects the base.

7. The median of a triangle divides it into two triangles of the same area.

8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.

II. Results on Quadrilaterals:
1. The diagonals of a parallelogram bisect each other.

2. Each diagonal of a parallelogram divides it into triangles of the same area.

3. The diagonals of a rectangle are equal and bisect each other.

4. The diagonals of a square are equal and bisect each other at right angles.

5. The diagonals of a rhombus are unequal and bisect each other at right angles.

6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

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Average

Average:

Average = (Sum of observations/Number of observations)

Average Speed:

Suppose a man covers a certain distance at x kmph and an equal distance at y kmph.

Then, the average speed druing the whole journey is (2xy/x + y) kmph.

For example:Average of 1,2,3,4,5 then
                 
            1+2+3+4+5/5=3

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Clock

Important Formulas - Clock

1.Minute Spaces

The face or dial of clock is a circle whose circumference is divided into 60 equal parts, named minute spaces

2.Hour hand and minute hand

A clock has two hands. The smaller hand is called the hour hand or short hand and the larger one is called minute hand or long hand.

3. 55 min spaces are gained by minute hand (with respect to hour hand) in 60 min.

(In 60 minutes, hour hand will move 5 min spaces while the minute hand will move 60 min spaces. In effect the space gain of minute hand with respect to hour hand will be 60 - 5 = 55 minutes.)

4.Both the hands of a clock coincide once in every hour.

5.The hands of a clock are in the same straight line when they are coincident or opposite to each other.

6.When the two hands of a clock are at right angles, they are 15 minute spaces apart.

7.When the hands of a clock are in opposite directions, they are 30 minute spaces apart.

8.Angle traced by hour hand in 12 hrs = 360°

9.Angle traced by minute hand in 60 min. = 360°.

10.If a watch or a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast.

11.If a watch or a clock indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.

12.The hands of a clock will be in straight line but opposite in direction, 22 times in a day

13.The hands of a clock coincide 22 times in a day

14.The hands of a clock are straight 44 times in a day

15.The hands of a clock are at right angles 44 times in a day

16.The two hands of a clock will be together between H and (H+1)o'clock at(60H/11)min past H o'clock.

17.The two hands of a clock will be in the same straight line but not together between H and(H+1)o'clock at
      (5H-30)*12/11 min past H,when H > 6
      (5H-30)*12/11 min past H,when H < 6

18.Angle between Hands of a clock
     When the min hand is behind the hour hand,the angle between the two hands at M min past H o'clock
      =30(H-M/5)+M/2 degree
     When the min hand is ahead of the hour hand,the angle between the two hands at M min past H o'clock
      =30(M/5-H)-M/2 degree

19.The two hands of the clock will be at right angle between H and (H+1)o'clock at
     (5H +_15)12/11 min past Ho'clock

20.The min hand of a clock overtakes the hour hand at intervals of M min of correct time.The clock gains or       loss in a day by
     (720/11-M)(60*24/M)minutes

21.Between H and (H+1)oclock,the two hands of a clock are M minutes apart at(5H+-M)12/11minutes            past H o'clock

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Calendar

Important Formulas - Calendar

1)Odd Days

Number of days more than the complete weeks are called odd days in a given period

2)Leap Year

A leap year has 366 days.

In a leap year, the month of February has 29 days

Every year divisible by 4 is a leap year, if it is not a century.

Examples: 

1952, 2008, 1680 etc. are leap years.
1991, 2003 etc. are not leap years

Every 4th century is a leap year and no other century is a leap year.

Examples:

400, 800, 1200 etc. are leap years.
100, 200, 1900 etc. are not leap years

3)Ordinary Year

The year which is not a leap year is an ordinary year.

An ordinary year has 365 days

4)Counting odd days and Calculating the day of any particular date

 1 ordinary year ≡ 365 days ≡ (52 weeks + 1 day)
   Hence number of odd days in 1 ordinary year= 1.

1 leap year ≡ 366 days ≡ (52 weeks + 2 days)
Hence number of odd days in 1 leap year= 2.

100 years ≡ (76 ordinary years + 24 leap years )
                ≡ (76 x 1 + 24 x 2) odd days
                ≡ 124 odd days.
                ≡ (17 weeks + 5 days)
                ≡ 5 odd days.
                   Hence number of odd days in 100 years = 5.

Number of odd days in 200 years = (5 x 2) = 10 ≡ 3 odd days
Number of odd days in 300 years = (5 x 3) = 15 ≡ 1 odd days
Number of odd days in 400 years = (5 x 4 + 1) = 21 ≡ 0 odd days


Similarly, the number of odd days in all 4th centuries (400, 800, 1200 etc.) = 0
Mapping of the number of odd day to the day of the week


Number of Odd Days :0 1 2 3 4 5 6
Day of the week : Sunday Monday Tuesday Wednesday Thursday Friday Saturday

5)Last day of a century cannot be Tuesday or Thursday or Saturday.

6)For the calendars of two different years to be the same, the following conditions must be satisfied.

 a.Both years must be of the same type. i.e., both years must be ordinary years or both years must be leap         years.

 b.1st January of both the years must be the same day of the week.

Code Months:

Jan     -    0
Feb    -    3
Mar   -    3
April  -    6
May  -     1
June  -     2
July   -     6
August-    2
Sep   -      5
Oct    -     0
Nov   -     3
Dec   -      5

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Problems on Ages

Important Formulas on "Problems on Ages" :

1. If the current age is x, then n times the age is nx.

2. If the current age is x, then age n years later/hence = x + n.

3. If the current age is x, then age n years ago = x - n.

4. The ages in a ratio a : b will be ax and bx.

5. If the current age is x, then 1/n of the age isx/n .

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Percentage

Percentage

Concept of Percentage:

By a certain percent, we mean that many hundredths.

Thus, x percent means x hundredths, written as x%.

To express x% as a fraction: We have, x% =x/100 .

    Thus, 20% = 20/100=1/5 .

To express a/b as a percent: We have, a/b=(a/bx 100)%.

    Thus,1/4=(1/4x 100) %= 25%.

Percentage Increase/Decrease:

If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is:

(R/(100 + R)x 100)%

If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is:

(R/(100 - R)x 100)%

Results on Population:

Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

1. Population after n years = P (1 +R/100)n

2. Population n years ago =P/(1 +R/100)n

Results on Depreciation:

Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

1. Value of the machine after n years = P(1 -R/100)n

2. Value of the machine n years ago =P/(1 -R/100)n
n
3. If A is R% more than B, then B is less than A by(R/(100 + R)*100)%.

4. If A is R% less than B, then B is more than A by(R/(100 - R)x 100)%

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Partnership

Partnership:

When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.

Ratio of Divisions of Gains:

When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.

Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year:

(A's share of profit) : (B's share of profit) = x : y.

When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now gain or loss is divided in the ratio of these capitals.

Suppose A invests Rs. x for p months and B invests Rs. y for q months then,

(A's share of profit) : (B's share of profit)= xp : yq.

Working and Sleeping Partners:

A partner who manages the the business is known as a working partner and the one who simply invests the money is a sleeping partner

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Profit and Loss

IMPORTANT FACTS

Cost Price:

The price, at which an article is purchased, is called its cost price, abbreviated as C.P.

Selling Price:

The price, at which an article is sold, is called its selling prices, abbreviated as S.P.

Profit or Gain:

If S.P. is greater than C.P., the seller is said to have a profit or gain.

Loss:

If S.P. is less than C.P., the seller is said to have incurred a loss.

IMPORTANT FORMULAE

Gain = (S.P.) - (C.P.)

Loss = (C.P.) - (S.P.)

Loss or gain is always reckoned on C.P.

Gain Percentage: (Gain %)

    Gain % =(Gain x 100/C.P).

Loss Percentage: (Loss %)

    Loss % =((Loss x 100)/C.P.)

Selling Price: (S.P.)

    SP =((100 + Gain %) /100x C.P)

Selling Price: (S.P.)

    SP =((100 - Loss %)/100x C.P).

Cost Price: (C.P.)

    C.P. =(100/(100 + Gain %)x S.P).
Cost Price: (C.P.)

    C.P. =(100/(100 - Loss %)x S.P).

If an article is sold at a gain of say 35%, then S.P. = 135% of C.P.

If an article is sold at a loss of say, 35% then S.P. = 65% of C.P.

When a person sells two similar 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:

    Loss % =(Common Loss and Gain %/10)2=(x/10)2 .

If a trader professes to sell his goods at cost price, but uses false weights, then

    Gain % =(Error/(True Value) - (Error)x 100) %.

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Compound Interest

Compound Interest:
Let Principal = P, Rate = R% per annum, Time = n years.

When interest is compound Annually:

   Amount = P(1 +R/100)n

When interest is compounded Half-yearly:

    Amount = P(1 +(R/2)/100)2n

When interest is compounded Quarterly:

    Amount = P(1 +(R/4)/100)4n

When interest is compounded Annually but time is in fraction, say 3(2/5) years.5
    Amount = P(1 +R/100)3x(1 + 2/3R/100)

When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively.

    Then, Amount = P(1 +R1/100) (1 +R2/100)(1 + R3/100) .

Present worth of Rs. x due n years hence is given by:

Present Worth = x/(1 +R/100) .

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Simple Interest

Principal:
The money borrowed or lent out for a certain period is called the principal or the sum.

Interest:
Extra money paid for using other's money is called interest.

Simple Interest (S.I.):
If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest.
Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then

(i). Simple Intereest = (P x R x T/100)

(ii). P=(100 x S.I./R x T); R = (100 x S.I./P x T) and T =(100 x S.I./P x R) .

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Time and Work

Time and Work

1. Work from Days:

   If A can do a piece of work in n days, then A's 1 day's work =	1	.
   	n

2. Days from Work:

   If A's 1 day's work =	1	,	then A can finish the work in n days.
   	n

3. Ratio:
   If A is thrice as good a workman as B, then:
   Ratio of work done by A and B = 3 : 1.
   Ratio of times taken by A and B to finish a work = 1 : 3.

Speed, Time and Distance:

Speed=(Distance/Time), Time =(Distance/Speed), Distance = (Speed x Time).

km/hr to m/sec conversion:

x km/hr =(x x 5/18)m/sec.

m/sec to km/hr conversion:

x m/sec =(x x 18/5)km/hr.

If the ratio of the speeds of A and B is a : b, then the ratio of the

the times taken by then to cover the same distance is 1/a:1/b or b : a.

Suppose a man covers a certain distance at x km/hr and an equal distance at y km/hr. Then,

the average speed during the whole journey is(2xy/x + y)km/hr.

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Height and Distance

Height and Distance

Trigonometry: In a right angled  OAB, where BOA = ,

1.   sin  =	Perpendicular	=	AB	;
	Hypotenuse		OB

2.   cos  =	Base	=	OA	;
	Hypotenuse		OB

3.  tan  =	Perpendicular	=	AB	;
	Base		OA

4.  cosec  =	1	=	OB	;
	sin		AB

5.   sec  =	1	=	OB	;
	cos		OA

6.  cot  =	1	=	OA	;
	tan		AB

Values of T-ratios:

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Time and Distance

 Time and Distance

1. Formula of Speed

   Speed =(Distance/Time)


   So Formula of speed is, speed is equal to distance upon time. If you remember this formula then you will learn next two formula two.

2. Formula of Time 

   Time =(Distance/Speed)


   So Formula of time is, time is equal to distance upon speed.

3. Formula of Distance 

   Distance = (Speed * Time)

   So distance is simply speed into time. All three formulae that formula of speed, formula of time and formula of distance are interrelated.

4. Convert from kph(km/h) to mps(m/sec)
   For converting kph(kilometer per hour) to mps(meter per second) we use following formula

   xkm/hr=(x*5/18)m/sec


6. Convert from mps(m/sec) to kph(km/h)
   For converting mps(meter per second) to kph(kilometre per hour) we use following formula

   xm/sec=(x*18/5)km/h



7. If the ratio of the speeds of A and B is a : b, then the ratio of the

   the times taken by then to cover the same distance is
   1/a:1/b or b:a


8. Suppose a man covers a certain distance at x km/hr and an equal distance at y km/hr. Then,
   the average speed during the whole journey is
   (2xy/x+y)

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Aptitude And Reasoning Portion

Aptitude And Reasoning  

    Aptitude Portion :

 Problems on Trains  
Time and Distance 
Height and Distance
Time and Work
Simple Interest
Compound Interest
Profit and Loss
Partnership
Percentage
Problems on Ages
Calendar
Clock
Average
Area
Volume and Surface Area
Permutation and Combination
Numbers Problems on Numbers
Formulas on H.C.F and L.C.M
Decimal Fraction
Simplification
Square Root and Cube Root
Surds and Indices
Ratio and Proportion
Chain Rule Pipes and Cistern Boats and Streams
Alligation or Mixture Logarithm Races and Games
Stocks and Shares Probability True Discount
Banker's Discount Odd Man Out and Series
Logical Deduction

• Reasoning Portion:

Number Series
Number and Ranking
Alphabet test
Letter and Symbol Series

Blood Relations
Input and Output
Coding and Decoding
Analogy
Artificial Language
Matching Definitions 
Making Judgments 
Verbal Reasoning
Logical Problems 
Logical Games 
Analyzing Arguments
Strong and Weak Arguments
Statement and Assumption 
Course of Action 
Statement and Conclusion
Cause and Effect
Statement and Argument
Verbal Classification

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