MU-NLPC/Calc-aqua_rat huggingface dataset:
A clockwise rotation around point Z (that is, a rotation
in the direction of the arrow) transforms the shaded
quadrilateral to the unshaded quadrilateral. The angle
of rotation is approximately Options:
A) 180o
B) 270o
C) 360o
D) 45o
E) 135o

Solution: The measure of \PZQ formed by the bottom edge of the shaded   quadrilateral, ZQ, and this same edge after the rotation, ZP, is   approximately 90.   The transformation of the shaded quadrilateral to the unshaded   quadrilateral is a clockwise rotation around point Z through an an-   gle equal to the measure of re   ex angle PZQ.   The measure of \PZQ added to the measure of re   ex \PZQ is   equal to the measure of one complete rotation, or 360.   Therefore, the measure of re   ex angle PZQ is approximately   360o - 90o or 270o.   Thus the clockwise rotation around point Z is through an angle of   approximately 270o   correct answer 270o
Answer: B
An error 19% in excess is made while measuring the side of a square. Now What is the percentage of error in the calculated area of the square?
Choose the correct choice from the following options
A) 6.64%
B) 41.61%
C) 45.64%
D) 46.64%
E) 10.64%

Solution: Percentage error in calculated area   =(19+19+(19Ã—19)/100)%=41.61%   ANSWER:41.61%
Answer: B
The total of company C's assets in 1994 was 250% greater than the total in 1993, which in turn was 400% greater than the total in 1992. If the total of company C's assets in in 1992 was N dollars, which one of the following represents company C's assets in 1994:  Select the correct option:
A) 7N	B) 8N	C) 9N	D) 12.5N	E) 17.5N

Solution: Let's use actual numbers. Starting in 1992, let's say company C had $100 in assets.   In 1993, the total assets were 400% GREATER, which means 1992 PLUS 400% of 1992:   $100 + 4x$100 = $100 + $400 =$500   In 1994, the total assets were 250% GREATER than they were in 1993, which means 1993 PLUS 300% of 1993:   $500 + 2.5x$500 100 +  400= 
100 +  400 = 500
$500   In 1994, the total assets were 250% GREATER than they were in 1993, which means 1993 PLUS 300% of 1993:   $500 + 2.5x$500= $500 + $1250 =$1750   This is 17.5 times the 1992 number, so the correct answer is 17.5N.   17.5N500 +  1250= 
500 +  1250 = 1_750
$1750   This is 17.5 times the 1992 number, so the correct answer is 17.5N.   17.5N
Answer: E
Set S is the prime integers between 0 and 45. If three numbers are chosen randomly from set S, what is the probability that the sum of these three numbers is odd?  Choose the correct option:  A) 15/56  B) 3/8  C) 15/28  D) 5/8  E) 11/14

Solution: Total number of combinations=14C3=364...............(1)   Total number of combinations including 2 and two other numbers=13C2=78......................(2)   Therefore the difference ie. (1)-(2) is the set of 3 primes without 2=(364-78)= 
(364-78) = 286
286   So, probability=286/364=11/14   11/14
Answer: E
A drink vendor has 50 liters of Maaza, 144 liters of Pepsi and 368 liters of Sprite. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?  Choose the correct choice from the following options:
A) 135
B) 237
C) 281
D) 300
E) 380

Solution: The number of liters in each can = HCF of 50, 144 and 368 = 2 liters.   Number of cans of Maaza = 50/2= 
50/2 = 25
25   Number of cans of Pepsi= 144/2= 
144/2 = 72
72   Number of cans of Sprite= 368/2= 
368/2 = 184
184   The total number of cans required= 25 + 72 + 184= 
25 + 72 + 184 = 281
281 cans.   ANSWER:281
Answer: C
The sum of all two digit numbers divisible by 9 is
Choose the correct answer.
A) 910 B) 585 C) 785 D) 685 E) 885

Solution: Required numbers are 18,27 . . . . . 99   This is an A.P. in which a=18,d=9 and l=99.   Let the number of terms in it be n.Then t=99   So a+(n-1)d=99.   18+(n-1)*9=99,   18+9n-9=99   9+9n=99   9n=99-9= 
99-9 = 90
90/9= 
90/9 = 10
10.   Required sum=n/2(a+l)   =10/2(18+99)   =585   Answer is 585.
Answer: B
x + y = 19 , and x + 3y = 1. Find the value of x + 2y Choose the most appropriate option
A) 20
B) 18
C) 11
D) 10
E) 5

Solution: Add these two equations   2x + 4y = 20   Divide by 2 ( to Get x + 2y)   Answer will be D. 10
Answer: D
The toll T, in dollars, for a truck using a certain bridge is given by the formula T = 2.50 + 0.50( x − 2), where x is the number of axles on the truck. What is the toll for an 18-wheel truck that has 2 wheels on its front axle and 4 wheels on each of its other axles? Choices.
A) $ 2.50	B) $ 3.00	C) $ 3.50	D) $ 4.00	E) $ 5.00

Solution: Number of wheels in truck = 18   Number of wheels on its front axle =2   number of wheels remaining = 16   number of axles remaining axles = 16/4= 
16/4 = 4
4   Total number of axles= 5   T = 2.50 + 0.50( x − 2)   = 2.50 + .5*3   = 2.5 +1.5= 
2.5 +1.5 = 4
4 $   Answer $ 4.00
Answer: D
Four athletes, Wen-Wen, Mikaila, Brian, and Jamal, won gold medals in the Olympics. If Mikaila won 2 more than Brian did, Jamal won 6 less than Wen-Wen did, and Wen-Wen won 2 more than Mikaila did, how many more gold medals did Brian win than Jamal did?
Choose the correct choice from the following answers
A) 4 B) 6 C) 2 D) 1 E) 12

Solution: In this word problem, the first key is to turn the story into equations:   Mikaila won 2 more than Brian did: M = 2 + B   Jamal won 6 less than Wen-Wen did: J = W - 6   Wen-Wen won 2 more than Mikaila did: W = 2 + M   Then, in classic GMAT style, the problem doesn't ask you to solve for a single variable, but instead to solve for the difference between B and J (how many more gold medals did Brian win than Jamal did?). This means that your goal is to get the equations in terms of B and J so that you can solve for the expression B - J.   Taking the first equation and solving for B, you have B = M - 2.   Taking the second equation and solving for J you have J = W - 6, and then with the third equation you can replace W with 2 + M so that both B and J are in terms of M: J = (2 + M) - 6.   So if B = M - 2 and J = M - 4, then:   B - J = (M - 2) - (M - 4)   B - J = M - 2 - M + 4 (remove the parentheses by distributing the multiplication)   B - J = 2, so the answer is 2.
Answer: C
John and Mike enter into a partnership by investing $700 and $300 respectively. At the end of one year, they divided their profits such that a third of the profit is divided equally for the efforts they have put into the business and the remaining amount of profit is divided in the ratio of the investments they made in the business. If John received $800 more than Mike did, what was the profit made by their business in that year?  Choose one.
A) 46457 B) 465745 C) 3000 D) 3456 E) 35346

Solution: Say the profit was $x.   John share = x/6 (half of the third) + (x-x/3)*0.7   Mike's share = x/6 (half of the third) + (x-x/3)*0.3   Thus (x-x/3)*0.7-(x-x/3)*0.3=800 --> x=3000.   answer is 3000
Answer: C
Find the least number when successively divided by 2, 3 and 7 leaves remainders 1, 2 and 3 respectively? Choose one
A) 80  B) 85  C) 65  D) 70  E) 90

Solution: 2) 65 (32 3) 32 (10 7) 10 (1   64 30 7   -------- -------- - ----- ===> 65   1 2 3   ANSWER 65
Answer: C
In an election contested by two parties, Party E secured 2% of the total votes more than Party F. If party R got 147,000 votes, by how many votes did it lose the election?  Choose the correct choice from the following choices.
A) 6,000
B) 7,000
C) 8,000
D) 10,000
E) 12,000

Solution: E + F = 100 ---- (1)   E - F = 2 ---- (2)   Solving (1) and (2) E = 51% and F = 49%   49% of Total = 147000   Total = 300000   E = 153000   Difference = 153000 - 147000= 
153000 - 147000 = 6_000
6000   6,000nswer: 6,000
Answer: A
The weight of a hollow sphere is directly dependent on its surface area. The surface area of a sphere is 4π·R^2, where R is the radius of the sphere. If a hollow sphere of radius 0.15 cm made of a certain metal weighs 8 grams, a hollow sphere of radius 0.3 cm made of the same metal would weigh how many grams E?	Choose the correct choice:
A) E=16 B) E=32 C) E=64 D) 128 E) 512

Solution: Weight directly proportional to 4 pi r^2   Now, 4pi is constant, so, weight is directly proportional to r^2.   When radius = 0.15, weight = 8, so (0.15) ^ 2 proportional to 8;   (0.15)^2 * 4 proportional to 8*4, solving further   (0.15)^2 * 2 ^2 = (0.15*2)^2 = 0.3 ^ 2; so answer = 32 (E=32)
Answer: B
A dishonest dealer professes to sell goods at the cost price but uses a weight of 800 grams per kg, what is his percent?  Options
A) 28% B) 25% C) 55% D) 28% E) 55%

Solution: 800 --- 200   100 --- ? => 25%   Answer: 25%
Answer: B
How many 4x4x4 cubes could fit in a 9x13x16 box? Choose the correct choice from the following answers:
A) 22  B) 28  C) 29  D) 31  E) 32

Solution: The answer is C) 29. 9x13x16 gives the box an area of 1,872. If you divide 1,872 by 64 (the total of 4x4x4), you get 29.25. This means 29 full boxes will fit in the larger box.
Answer: C
The two lines y = x and x = -4 intersect on the coordinate plane. If z represents the area of the figure formed by the intersecting lines and the x-axis, what is the side length T of a cube whose surface area is equal to 6z? Select.
A) T=16
B) T=8√2
C) T=8
D) T=2√2
E) (√2)/3

Solution: 800score Official Solution:   The first step to solving this problem is to actually graph the two lines. The lines intersect at the point (-4, -4) and form a right triangle whose base length and height are both equal to 4. As you know, the area of a triangle is equal to one half the product of its base length and height: A = (1/2)bh = (1/2)(4 × 4) = 8; so z = 8.   The next step requires us to find the length of a side of a cube that has a face area equal to 8. As you know the 6 faces of a cube are squares. So, we can reduce the problem to finding the length of the side of a square that has an area of 8. Since the area of a square is equal to s², where s is the length of one of its side, we can write and solve the equation s² = 8. Clearly s = √8 = 2√2 , oranswer choice (T=2√2).
Answer: D
Which number is the odd one out?
571219
461016
831114
461016
971613
781523 Choose the correct choice from the following options: A) 971533 B) 971613 C) 641613 D) 851613 E) 931613

Solution: B   971613   Numbers are obtained by adding pairs of digits, ie with 571219.   5 + 7= 
5 + 7 = 12
12; 7 + 12= 19.   To follow this same pattern 971613 would have to be 971623.
Answer: B
A positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. If the six-digit integer is divisible by 3, and n is of the form 12k,k24, where k represents a digit that occurs twice, how many values could n have?
Choose the correct choice from the following answers:
A) 2
B) 3
C) 6
D) 5
E) 4

Solution: 12kk24   Taking sum of the numericals = 1 + 2 + 2 + 4= 
1 + 2 + 2 + 4 = 9
9   We require the values of K such that they are also divisible by 3   120024   123324   126624   129924   Answer= 4 = 4
Answer: E
If x ≥ 0, y ≥ 0 and (x + y) ≤ 1, then the maximum value of (2x + 3y) is	Choose the most appropriate option.
A) 2	B) 3	C) 4	D) 5	E) 6

Solution: Sol.(b) It is given that x ≥ 0, y ≥ 0 and (x + y) ≤ 1.   x + y ≤ 1 => 2(x + y) ≤ 2 => 2x + 2y ≤ 2.   => 2x + 2y + y ≤ 2 + y   => 2x + 3y ≤  2 + 1= 
 2 + 1 = 3
3. (since y ≤ 1).   Answer 3
Answer: B
Each of the cucumbers in 100 pounds of cucumbers is composed of 99% water, by weight. After some of the water evaporates, the cucumbers are now 95% water by weight. What is the new weight of the cucumbers, in pounds?
Pick:
A) 2
B) 20
C) 92
D) 96
E) 98

Solution: Out of 100 pounds 99% or 99 pounds is water and 1 pound is non-water. After somewaterevaporates the cucumbers become 95% water and 5% of non-water, so now 1 pound of non-water composes 5% of cucucmbers, which means that the new weight of cucumbers is  1/0.05= 
 1/0.05 = 20
20 pounds.   Answer: 20.
Answer: B
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the probability Q that the 2 apples selected will include the spoiled one?  Choose the correct choice from the following answers:
A) a. 1/5 B) b. 3/10 C) c. 2/5 D) d. 1/2 E) e. 3/5

Solution: there can be two ways in which one can select a spoiled apple and a good apple.   1. you pick the spoiled one first and the good one later..   Prob of picking bad one = 1/5 (one bad apple)   Prob of picking a good one from the remaining 4 good apples (4/4 =1)   therefore the total prob for case 1 = i/5 * 1 = 1/5   Total prob Q= porb1 + prob2 = 1/5+1/5 = 2/5.c. 2/5
Answer: C
A tank is filled to one quarter of its capacity with a mixture consisting of water and sodium chloride. The proportion of sodium chloride in the tank is 50% by volume and the capacity of the tank is 24 gallons. If the water evaporates from the tank at the rate of 0.5 gallons per hour, and the amount of sodium chloride stays the same, what will be the concentration of water in the mixture in 4 hours? Select the correct option: A) 40% B) 35% C) 30% D) 25% E) 20%

Solution: The number of gallons in the tank is (1/4)24 = 6 gallons   The amount of sodium chloride is 0.5(6) = 3 gallons   At the start, the amount of water is 0.5(6) = 3 gallons   After 4 hours, the amount of water is 3 - 0.5(4) = 1 gallon   The concentration of water is 1 / (3+1) = 1/4 = 25%   The answer is 25%.
Answer: D
A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is?
Choose the correct choice from the following options:
A) 28 B) 50 C) 28 D) 27 E) 21

Solution: Speed of the train relative to man = (125/10) m/sec = (25/2) m/sec. [(25/2) * (18/5)] km/hr   = 45 km/hr. Let the speed of the train be x km/hr. Then, relative speed   = (x - 5) km/hr. x - 5 = 45 ==> x   = 50 km/hr.   Answer: 50
Answer: B
The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 90%, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?	Options.
A) 100% decrease B) 50% decrease C) 40% decrease D) 40% increase E) 30% increase

Solution: The rate of reaction is invesrsly proportional to the concentration of chemical B present. It used to have B=1 . Now that B is increased by 100%. So the new equation would be 2B=(1/2). In order for the rate of reaction to still be 1, we need to change the concentration of A to yield a 2. It used to be A^2=1, now the new equation should be (sqrt(2)*A)^2=2. The change in the concentration of A can be calculated as (sqrt(2) -1)/1 or approximately 40% increase. Answer 30% increase.
Answer: E
Three is the largest number that can be divided evenly into 27 and the positive integer x, while 10 is the largest number that can be divided evenly into both 100 and x. Which of the following is the largest possible number that could be divided into x and 2500  Choose the correct option.
A) 30
B) 50
C) 210
D) 300
E) 700

Solution: 27 = 3*3*3, and its GCF with x is 3, which implies that x has ONE factor of 3, but not TWO factors of 3.   100 and x have a GCF of 10, which implies that x has ONE factor of 10, but not TWO factors of 10.   Then we want to know what is the largest possible GCF of x and 2100.   Well 2500 = 5 * 5 * 10 * 10   We want x to include as many factors in common with 2100 as possible, to make the GCF with 2100 as big as possible.   We know x has one factor of 3, but not two factors --- that takes the 3.   We know x has one factor of 10, but not two factors --- we can take one of those 10's, but we have to leave the other   No other restrictions, so we can also grab that factor of 7 --- nothing saying that we can't, and it's there for the taking.   5*10= 
5*10 = 50
50   If we allow x to include as many factors as possible, within the constraints given, that is the most x could have in common with 2100.   50
Answer: B
The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of R is?	Choose the correct option.
A) A)Rs.4078	B) B)Rs.4000	C) C)Rs.4029	D) D)Rs.4027	E) E)Rs.6400

Solution: Let P, Q and R represent their respective monthly incomes. Then, we have:   P + Q = (5050 * 2)= 
(5050 * 2) = 10_100
10100 --- (i)   Q + R= (6250 * 2)= 
(6250 * 2) = 12_500
12500 --- (ii)   P + R= (5200 * 2)= 
(5200 * 2) = 10_400
10400 --- (iii)   Adding (i), (ii) and (iii), we get:   2(P + Q + R)= 33000 = P + Q + R = 16500 --- (iv)   Subtracting (i) from (iv), we get, P = 6400.   P's monthly income = Rs.6400.   Answer: Rs.6400
Answer: E
A restaurant spends one fifth of its monthly budget for rent and half of the rest for food and beverages. What percentage of the budget does the restaurant spend for food and beverages?
Pick:
A) 23.5% B) 32.5% C) 35% D) 37.5% E) 40%

Solution: Spend on rent = 1/5   Spend on food and beverage= 1/2 of remaining = 1/2*4/5= 4/10=2/5   so 2/5= 40.0%   40% is the answer
Answer: E
When positive integer d is divided by 6, the remainder is 1. Which of the following must be true?
I. d is a prime number
II. d is odd
III. d is divisible by 7 Choose the correct answer:	A) I only	B) II only	C) III only	D) I and II only	E) II and III only

Solution: d=6q+1   Examine II) put q=0,1,2,3....... we have d=1,7,13,...25 so d must be odd   Examine III) 14/6 has reminder 249/7 has reminder 1 so not always true   Examine I) d= 7 true but while 2 is prime, it does not give reminder 1   Answer: II only
Answer: B
How many even integers from 1 to 100 (both inclusive) have EVEN number of factors?  Pick one:
A) 13 B) 14 C) 5 D) 6 E) 7

Solution: Integers having even number of factors will be perfect squares. even numbers will have even perfect squares. Thus, the possible values for the perfect squares are :   4,16,36,64,100 and the corresponding integers are 2,4,6,8,10(more than 4). Thus 5 is the correct answer .
Answer: C
John must arrange 2 different physics books and 3 different mathematics books on one shelf. If the first book on the leftmost side is a mathematics book and no physics book is next to another physics book. How many different arrangements exist?	Pick.
A) 6 B) 12 C) 36 D) 120 E) 720

Solution: First Place, is reserved for Math book. There are 3 Math books which can be arranged in3 ways   Restriction comes into playhysics book cannot be together. Hence, the maththe physics books take alternate places   Second Place, Physics Book - in 3 ways   Third Place, Math Book - in 2 ways (only 2 books left)   Fourth Place, Physics - in 2 ways (only 2 Physics book left)   Rest two places can be arranged in one way only.   Total no of ways = 3*2*2*1*1= 
3*2*2*1*1 = 12
12 ways   12
Answer: B
If a month of the year is chosen randomly, how likely is it that it contains 5 or fewer letters, or that the month starts with a "J"?	Select:	A) 7/12	B) 1/2	C) 2/3	D) 3/4	E) 11/12

Solution: P(5á´œJ) = P(5) + P(J) - P(5âˆ©J), where 5 denotes "five or fewer letters", and J denotes "starts with a J".   P(5á´œJ) = 5/12 + 3/12 - 2/12   = 1/2   Answer: 1/2
Answer: B
Find the number of ways of arranging a girl and 20 boys at a circular table so that the girl always sits in a particular seat? Options
A) 3!	B) 8!	C) 11!	D) 12!	E) 20!

Solution: Ans.(E)   Sol. Total number of persons = 21 Girl can sit in a particular seat in one way. Now, remaining positions are defined relative to the girl. Hence, the remaining can sit in 20 places => 20! Ways ... The number of required arrangements = 20! x 1= 20! = 20! ways
Answer: E
An even and an odd integer are multiplied together. Which of the following numbers could not be the square of their product?  Choices:
A) 36 B) 81 C) 100 D) 324 E) 900

Solution: 81 = 9^2 so it can not be the square of an even number multiplied by an odd number.   The answer is 81.
Answer: B
The radius of a cylinder is 6 m, height 21 m. The total surface area of the cylinder is?	Select the correct option:
A) 525
B) 528
C) 522
D) 529
E) 521

Solution: Total surface area of the cylinder is = 2πr (r+h)   = 2 × 22/7 × 6(6+8)   = 2 × 22/7 × 6(14) = 44 × 12 = 528 m(power2)   Answer is 528.
Answer: B
What will come in place of the x in the following Number series? 40, 160, x, 2560, 10240	Choose the correct choice from the following options:
A) 460
B) 200
C) 900
D) 800
E) 640

Solution: (640)   10 x 4 = 40, 40 x 4 = 160, 160 x 4 = 640, 640 x 4 = 2560.
Answer: E
There r 4 prime numbers in ascending order. The multiplication of first 3 is 385 and that of last 3is 1001. what is the last number is Select.
A) 11 B) 12 C) 14 D) 13 E) 15

Solution: Explanation:   abcbcd=3851001=>ad=513   abcbcd=3851001=>ad=513   So d = 13   13
Answer: D
A frame 2 inches wide is placed around a rectangular picture with dimensions 8 inches by 12 inches. What is the area of the frame, in square inches?  Choose the correct choice from the following.
A) 44	B) 96	C) 128	D) 144	E) 168

Solution: A frame 2 inches wide is placed around a rectangular picture with dimensions 8 inches by 12 inches. What is the area of the frame, in square inches?   (12*16)-(8*12)= 
(12*16)-(8*12) = 96
96   ANSWER:96
Answer: B
How many liters of oil at Rs.40 per liter should be mixed with 240 liters of a second variety of oil at Rs.60 per liter so as to get a mixture whose cost is Rs.52 per liter?	Choose the correct choice from the following options.
A) 120 liters  B) 180 liters  C) 110 liters  D) 160 liters  E) 170 liters

Solution: Explanation:   =8:12=2:3   ANSWER IS 160 liters
Answer: D
A train 300 m long is running at a speed of 45 km/hr. In what time will it pass a bridge 150 m long?  Select the correct option
A) 40  B) 36  C) 41  D) 42  E) 34

Solution: Speed = 45 * 5/18 = 25/2 m/sec   Total distance covered = 300 + 150= 
300 + 150 = 450
450 m   Required time= 450 * 2/25= 
450 * 2/25 = 36
36 sec   Answer: Option 36
Answer: B
The maximum numbers of students among them 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is Choose the correct choice:
A) 91
B) 910
C) 1001
D) 1911
E) None

Solution: olution   Required number of students.   = H.C.F of 1001 and 910.   ‹=› 91.   91nswer 91
Answer: A
What is the units digit of 26! + 50! + 4! + 4!? Choices.
A) 8	B) 6	C) 4	D) 2	E) 0

Solution: Let us first write down the factorials of different numbers:   1! = 1   2! = 2   3! = 6   4! = 24   5! = 120   6! = 720   and so on.   If we notice here, all the numbers after 4! will end with a 0.   Here is why:   If we multiply any number by 10, a 0 is added to that number   And a 10 is formed by a 5 and a 2, 10 = 5*2   Till 4!, we do not have a "5" in multiplication. But after that, we will have atleast one "5" in the multiplication.   Coming back to the question,   Units digit of 26! + 50! + 4! + 4! = ?   By the logic above, 26! and 50! will end in a "0"   4! will end in a 4   Hence the units digit = 0 + 0 + 4 + 4= 
0 + 0 + 4 + 4 = 8
8   Correct Option: 8
Answer: A
There are 13 South American countries, Annie has won a ticket to fly from Denver to one South American country and then directly to another South American Country before returning home to Denver again. Assuming there are four airports in each country, how many flights paths can Annie choose from?
Choices
A) 13.12.11.10.4.4.	B) 13.12.11.10.4	C) 13.12.4.4.	D) 13.12.4	E) 13.13.13.13

Solution: flights paths can Annie choose from There are 13 ways to chose the first destination,   then, 12 ways to chose next destination, with option of 4 airports (13x4)   then, to return he has 4 options to chose the airports   Thus, 13*12*4*4 ways.   Using permutations:   If the solution had written 13P1 instead of 13C1 it won't matter. As per the definition - making arrangements of the combinations gives permutations.   Since there are no arrangement for 1 item, permutationscombinations are equal.13.12.4.4.
Answer: C
In the class of 50 students, 30 speak Tamil and 30 speak Telugu. What is the lowest possible number of students who
speak both the languages? Choose the correct option.
A) A)8  B) B)10  C) C)20  D) D)30  E) E)32

Solution: let the student who speaks tamil-x   let the student who speaks telugu-y   as (xuy)-(xny)=total   30+30-(xny)=50   =10   10)
Answer: B