Permutation and Combination- Aptitude Questions and Answers

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In how many different ways can the letters of the word ‘DESIGN’ be arranged so that the vowels are at the two ends?

A)

56


B)

64


C)

24


D)

48



Correct Answer :

48


Explanation :

DESIGN has 2 vowels(EI) and 4 consonants (DSGN).
Four consonants can be arranged in 4! ways two vowels can be arranged in 2! ways

Total number of arrangements= 4!×2! = 48

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By using digits 1 to 9 how many 4 digit numbers can be formed which has 5 as a unit digit? Repetition of digits is are not allowed.

A)
336

B)
1400

C)
504

D)
251


Correct Answer :

336


Explanation :

5 at the unit place or divisible by 5. So, there is 1 way of doing it.
No repetition of digits.thus there are 8 ways of filling the tens place from the remaining 8 digits (1,2,3,4,6,7,8,9).
the hundreds place can be filled by any of the remaining 7 digits. So, there are 7 ways of filling it.
the thousands place can be filled by any of the remaining 6 digits. So, there are 6 ways of filling it.

Required number of numbers = (1 * 8 * 7 * 6) = 336.

In how many ways can 5 prizes be given away to 7 boys when each boy is eligible for all the prizes?

A)
5!

B)
7!

C)
57

D)
75


Correct Answer :

75


Explanation :

A prize can be given to any one of the 7 boys in (7/1)=7 ways
So, each prize can be distributed in (7/1)=7 ways
5 prizes can be distributed in (7/1)*(7/1)*(7/1)*(7/1)*(7/1) = 75 ways

How many different words, can be formed by rearranging the letters of the word ‘SUCCESS’?

A)
5040

B)
840

C)
420

D)
1680


Correct Answer :

420


Explanation :

The word "SUCCESS" has total 7 elements, there two repeated elements S(thrice) and C(twice). The formula for permutations with repeated elements is as follows, When k out of n elements are indistinguishable, e.g. k copies of the same book, the number of different permutations is n!/k!.

Therefore the total number of arrangement of the word SUCCESS can be formed is 7!/3! *2!
=> (7*6*5*4*3*2*1)/(3*2*1)*(2*1) => 7*6*5*2 => 420

In how many different ways can the letters of the word ‘ARCHIVE’ be arranged?

A)
2630

B)
5040

C)
1680

D)
3456


Correct Answer :

5040


Explanation :

There are 7 unique letters in ARCHIVE which means that there are 7!=7*6*5*4*3*2*1=5040 different ways to rearrange them.

In how many different ways can the letters of the word ‘PUBLIC’ be arranged so that the vowels are at the two ends?

A)
48

B)
72

C)
36

D)
None of these


Correct Answer :

48


Explanation :

PUBLIC has 2 vowels(UI) and 4 consonants (PBLC). All the letters are distinct.
First letter can be any of the two vowels (2 ways) and Last letter is the remaining vowel (1 way)
4 consonants can be arranged in 4! ways

Total number of ways = 2!*1!*4! = (2*1)*1*(4*3*2*1)=48

If 56Pr + 6 : 54Pr + 3 = 30800 : 1 then the value of r is ?

A) 40
B) 41
C) 42
D) 43

Correct Answer : 41
Explanation :

56Pr + 6 : 54Pr + 3 = 30800 : 1
⇒ 56! / (56 -(r + 6))! : 54! / (54 - (r + 3))! = 30800 : 1
⇒ 56! / (50 - r)! : 54! / (51 - r)! = 30800 : 1

As 56! = 56*55*54!
and also (51 - r)! =(51 - r)*[(51 - r) -1]! ⇒ (51 - r)*(50 - r)!

So,
⇒ 56*55*54! / (50 - r)! = (30800 x 54!) / (51 - r)*(50 - r)!
⇒ 56*55 = 30800/(51 - r)
⇒ 51 - r = 10
Hence, r = 41

How many groups of 6 persons can be formed from 8 men and 7 women ?

A) 5000
B) 5005
C) 5050
D) 5650

Correct Answer : 5005
Explanation :

Total number of person = 8 men + 7 women = 15
No. of groups can be formed of 6 persons= 15C6 = 15! / {6! *(15 - 6)!} = 15! / (6! * 9!)
= (15 x 14 x 13 x 12 x 11 x 10) / (6 x 5 x 4 x 3 x 2 x 1)
= 5005

If 10 persons meet at a meeting and shake hands exactly once with each other, what is the total number of hand shakes?

A)
35

B)
35

C)
50

D)
45


Correct Answer :

45


Explanation :

Combinations of n objects taken r at a time = nCr= n! / [r!(n-r!)]
Required number of ways = 10C2 = 10*9 / 2 = 45

In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is?

A)
200

B)
216

C)
235

D)
256


Correct Answer :

200


Explanation :

Let there were g girls and b boys.
As given, total number of games in which both players were women gC2 = 45
=> g(g−1)/2=45
=> g2-g-90 = 0
(g - 10)(g + 9) = 0
So, g =10 as g ≠ -9

As given, total number of games in which both players were men bC2 = 190
=>b(b−1)/2=190
=> b2-b-380 = 0
(b - 20)(b + 190) = 0
So, b=20 as b ≠ -190

Total number of games= (g + b)C2 => 30C2 => (30 *29) / (2* 1) => 435

Number of games in which one player is a boy and the other is a girl = 435 - 45 - 190 = 200

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Pipes and Cistern

Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A s opened first, in how many hours, the tank shall be full?

A large tanker can be filled by two pipes A and B in 60 min and 40 min respectively. How many minutes will it take to fill the tanker from empty state if B is used for first half of the time, and A and B fill it together for the other half?

A tank is filled by 3 pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The 2nd pipe fills the tank 5 hours faster than first pipe and 4 hours slower than third pipe. The time required by first pipe is?

Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

Two pipes A and B can fill a tank in 24 min and 32 min respectively. If both the pipes are opened simultaneously, after how much time B should be closed so that the tank is full in 18 min?

Two pipes A and B can fill a tank in 36 min and 45 min respectively. A water pipe C can empty the tank in 30 min. First A and B are opened. after 7 min, C is also opened. In how much time, the tank is full?

Two pipes can fill a cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 min more to fill the cistern. When the cistern is full, in what time will the leak empty it?

Bucket P has thrice the capacity as Bucket Q. It takes 60 turns for Bucket P to fill the empty drum. How many turns it will take for both the buckets P and Q, having each turn together to fill the empty drum?

Probability

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In a lottery ,there are 10 prizes and 25 blanks.A lottery is drawn at random. what is the probability of getting a prize ?

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Two diced are tossed the probability that the total score is a prime number?

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An unbiased die is tossed.Find the probability of getting a multiple of 3?

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