Permutations & Combinations Mcqs - Set 1

1)   A question paper consists of five problems, each problem having three internal choices. In how many ways can a candidate attempt one or more problems?

a. 63
b. 511
c. 1023
d. 15
Answer  Explanation 

ANSWER: 1023

Explanation:
Given that, the question paper consists of five problems. For each problem, one or two or three or none of the choices can be attempted.

Hence, the required number of ways = 45 – 1.
= 210 – 1 = 1024 – 1 = 1023


2)   In a class there are 20 boys and 25 girls. In how many ways can a boy and a girl be selected?

a. 400
b. 500
c. 600
d. 20
Answer  Explanation 

ANSWER: 500

Explanation:
We can select one boy from 20 boys in 20 ways.

We select one girl from 25 girls in 25 ways
We select a boy and girl in 20 * 25 ways i.e., = 500 ways.


3)   In how many ways can three consonants and two vowels be selected from the letters of the word “TRIANGLE”?

a. 25
b. 13
c. 40
d. 30
Answer  Explanation 

ANSWER: 30

Explanation:
The word contains five consonants. Three vowels, three consonants can be selected from five consonants in ⁵C₃ ways, two vowels can be selected from three vowels in ³C₂ ways.

3 consonants and 2 vowels can be selected in ⁵C₂ . ³C₂ ways i.e., 10 * 3 = 30 ways.


4)   A bag contains nine yellow balls, three white balls and four red balls. In how many ways can two balls be drawn from the bag?

a. ⁹C₂
b. ³C₂
c. ¹⁶C₂
d. ¹²C₂
Answer  Explanation 

ANSWER: ¹⁶C₂

Explanation:
Total number of balls = 9 + 3 + 4

Two balls can be drawn from 16 balls in ¹⁶C₂ ways.


5)   How many four digit even numbers can be formed using the digits {2, 3, 5, 1, 7, 9}

a. 60
b. 360
c. 120
d. 240
Answer  Explanation 

ANSWER: 60

Explanation:
The given digits are 1, 2, 3, 5, 7, 9

A number is even when its units digit is even. Of the given digits, two is the only even digit.
Units place is filled with only ‘2’ and the remaining three places can be filled in ⁵P₃ ways.
Number of even numbers = ⁵P₃ = 60.


6)   How many four digit numbers can be formed using the digits {1, 3, 4, 5, 7,9}(repetition of digits is not allowed)?

a. 360
b. 60
c. 300
d. 180
Answer  Explanation 

ANSWER: 360

Explanation:
The given digits are six.

The number of four digit numbers that can be formed using six digits is ⁶P₄ = 6 * 5 * 4 * 3 = 360.


7)   The number of ways in which six boys and six girls can be seated in a row for a photograph so that no two girls sit together is_________?

a. (6!)2
b. 6! * ⁷P₆
c. 2(6!)
d. 6! * 7
Answer  Explanation 

ANSWER: 6! * ⁷P₆

Explanation:
We can initially arrange the six boys in 6! ways.

Having done this, now three are seven places and six girls to be arranged. This can be done in ⁷P₆ ways.
Hence required number of ways = 6! * ⁷P₆


8)   A letter lock consists of three rings each marked with six different letters. The number of distinct unsuccessful attempts to open the lock is at the most__________?

a. 216
b. 243
c. 215
d. 729
Answer  Explanation 

ANSWER: 215

Explanation:
Since each ring consists of six different letters, the total number of attempts possible with the three rings is = 6 * 6 * 6 = 216. Of these attempts, one of them is a successful attempt.

Maximum number of unsuccessful attempts = 216 – 1 = 215.


9)   A committee has 5 men and 6 women. What are the number of ways of selecting a group of eight persons?

a. 165
b. 185
c. 205
d. 225
Answer  Explanation 

ANSWER: 165

Explanation:
Total number of persons in the committee = 5 + 6 = 11

Number of ways of selecting group of eight persons = ¹¹C₈ = ¹¹C₃ = (11 * 10 * 9)/(3 * 2) = 165 ways.


10)   What are the number of ways to select 3 men and 2 women such that one man and one woman are always selected?

a. 100
b. 60
c. 30
d. 20
Answer  Explanation 

ANSWER: 30

Explanation:
The number of ways to select three men and two women such that one man and one woman are always selected = Number of ways selecting two men and one woman from men and five women

= ⁴C₂ * ⁵C₁ = (4 * 3)/(2 * 1) * 5
= 30 ways.