Permutations and combinations - Quantitative Aptitude (MCQ) questions

1)   There are 30 people in a party. If everyone is to shake hands with one another, how many hand shakes are possible?
- Published on 06 May 16

a. 180
b. 256
c. 386
d. 435
Answer  Explanation 

ANSWER: 435

Explanation:
Total number of persons = n = 30
Shakehands involve only 2 persons = r = 2
Number of shakehands = nCr = 30C2
30C2 = (30 * 29) /(2 * 1) = 435
nCr = (n!) / r! (n – r)!
= 30! / 2! 28!
= 435


2)   A box contains 4 black, 3 red and 6 green marbles. 2 marbles are drawn from the box at random. What is the probability that both the marbles are of the same color?
- Published on 04 May 16

a. 12/74
b. 24/78
c. 13/78
d. None of these
Answer  Explanation 

ANSWER: 24/78

Explanation:
Total marbles in a box = 4 black + 3 red + 6 green marbles = 13 marbles
2 marbles are drawn from 13 marbles at random. Therefore,
n(S) = 13C2 = 78 ways
Let A be the event that 2 marbles drawn at random are of the same color. Number of cases favorable to the event A is
n(A) = 4C2 + 3C2 + 6C2 = 6 +3 + 15 = 24
Therefore, by definition of probability of event A,
P(A) = n(A)/n(S) = 24/78


3)   A box contains 2 white, 3 black and 5 red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
- Published on 19 Apr 16

a. 29
b. 36
c. 48
d. 85
e. None of these
Answer  Explanation 

ANSWER: 85

Explanation:
Total balls in a box = 2W + 3B + 5R = 10 balls.
Therefore total number of ways of drawing 3 balls from 10 balls is 10C3 = 120 ways ........(1)
This includes all types of color combinations of white, black and red.
Now total balls in a box of white and red color = 2W + 5R = 7 balls.
Therefore total number of ways of drawing 3 balls from 7 balls is 7C3 = 35 ways ........(2)
This includes all types of color combinations of white and red only.
Therefore from (1) and (2), we get the total number of ways of drawing 3 balls, which includes at least one black ball = 120 – 35 = 85 ways.


4)   On a shelf, 2 books of Geology, 2 books of Sociology and 5 of Economics are to be arranged in such a way that the books of any subject are to be together. Find in how many ways can this be done?
- Published on 01 Apr 16

a. 3846
b. 2880
c. 900
d. 90
e. None of these
Answer  Explanation 

ANSWER: 2880

Explanation:
There are books of 3 subjects (Geology, Sociology and Economics), hence they can be arranged in 3! (3 * 2 * 1) = 6 ways.
Further, in each category (subject), books are to be arranged in different order, we get,
Required number of ways: 3! * [2! * 2! * 5!] = 2880


5)   A briefcase has a number-lock system containing a combination of 3 digits (Each digit can be of numbers 0 to 8). If the correct combination is unknown, how much maximum time would be required to open the bag if each “trial” of combination takes 3 seconds?
- Published on 01 Apr 16

a. 45.23 minutes
b. 36.45 minutes
c. 60.34 minutes
d. 90.15 minutes
e. 50.9 minutes
Answer  Explanation 

ANSWER: 36.45 minutes

Explanation:
Maximum number of trials required = 9 * 9 * 9 = 729. Since for each combination trial, 3 seconds are required to open the briefcase is given as 3 * 729 = 2187 seconds = 36.45 minutes.


6)   A person can go from place “P” to “Q” by 7 different modes of transport, but is allowed to return back to “P” by any mode other than the one used earlier. In how many different ways can he complete the entire journey?
- Published on 01 Apr 16

a. 42
b. 30
c. 11
d. 56
e. 65
Answer  Explanation 

ANSWER: 42

Explanation:
The person can travel from “P” to “Q” in any of the 7 different modes of transport. However, while returning, he cannot use the mode which he used earlier. Thus, while returning, he has the option of only 6 different modes of transport. Hence, the total number of ways would be 7 * 6 = 42.


7)   How many words can be formed by using all letters of word ALIVE.
- Published on 25 Mar 16

a. 86
b. 95
c. 105
d. 120
Answer  Explanation 

ANSWER: 120

Explanation:
The word ALIVE contains 5 different letters
Therefore,
Required number of words = 5p5 = 5!
= (5*4*3*2*1) = 120


8)   How many 3-letter words can be formed out of the letters of the word ‘CORPORATION’, if repetition of letters is not allowed?
- Published on 06 Jul 15

a. 990
b. 336
c. 720
d. 504
Answer  Explanation 

ANSWER: 336

Explanation:
There are in all 11 letters in the word ‘CORPORATION’. Since repetition is not allowed, there are 8 different letters that can be used to form 3-letter word.

Therefore, total number of words that can be formed = 8P3 = (8*7*6)

= 336


9)   In how many different ways can the letters of the word ‘GEOMETRY’ be arranged so that the vowels always come together?
- Published on 06 Jul 15

a. 720
b. 4320
c. 2160
d. 40320
Answer  Explanation 

ANSWER: 4320

Explanation:

There are in all 8 letters in the given word of which 3 are vowels. As the vowels should always be together, considering the 3 vowels as one letter, there are in all 6

letters which can be arranged in 6! ways = 720
Also the 3 vowels can be arranged in 3! ways = 6
Total number of arrangements = 720*6 = 4320


10)   In a group containing 6 cows and 4 buffalos, 4 livestock are to be selected in such a way that at least 1 cow should always be present. How many way of doing that are possible?
- Published on 15 Jun 15

a. 209
b. 205
c. 194
d. 163
Answer  Explanation 

ANSWER: 209

Explanation:
No explanation is available for this question!


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