Permutations & Combinations Mcqs - Set 3

1)   In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

a. 63
b. 90
c. 126
d. 45
Answer  Explanation 

ANSWER: 63

Explanation:
No explanation is available for this question!

2)   In how many different ways can four books A, B, C and D be arranged one above another in a vertical order such that the books A and B are never in continuous position?

a. 9
b. 12
c. 14
d. 18
Answer  Explanation 

ANSWER: 12

Explanation:
The number of arrangement in which A and B are not together

= Total number of arrangements

= Number of arrangements in which A and B are together =4!-3!x2! = 24-12 =12.

3)   There are two identical red, two identical black, and two identical white balls.In how many different ways can the balls be placed in the cells (Each cell to contain one ball) shown above such that balls of the same colour do not occupy any two consecutive cells?

a. 15
b. 18
c. 24
d. 30
Answer  Explanation 

ANSWER: 24

Explanation:
Case I : 2 balls of the same colour and two balls are a different colour are arranged.

Two balls of the same colour and two balls of different colours can be arranged together in which two balls of the same colour are adjacent =4!/2!x2! = 6 ways

Therefore, Total number of arrangements = 6×3 =18 ways

Case II : Two colours out of 3 can be selected in = 3C1 = 3ways

Now 2 balls of each colour can be arranged alternatively in 2 ways

Thus 4 balls can be arranged(two of each colours)

= 3×2 = 6ways

Hence total number of arrangements = 18+6 =24 ways

4)   Twelve people from a club, by picking lots. One of them will host a dinner for all once in a month. The number of dinners a particular member has to host in one year is________?

a. One
b. Zero
c. Three
d. Cannot be determined
Answer  Explanation 

ANSWER: Cannot be determined

Explanation:
We cannot predicted the number of dinners that the particular number has to host in one year.

5)   A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair while preparing the bill the bill clerk inter-changed the number of black and brown pairs by mistake which increased the bill by 100% what was the number of pair of brown socks in the original order?

a. 10
b. 15
c. 20
d. 25
Answer  Explanation 

ANSWER: 25

Explanation:
Let the bought x pairs of brown socks and the price of each brown pair be Y.

Then total cost = 5x3Y+xy

Changed cost = 5xY+x*3Y

According to the question .

5y+3xy –(15y+xy)/15y+xy x 100% = 100%

=> 5y+3xy-15y-xy/ 15y+xy x100% = 100%

=> 5y+3xy = 15y+xy+15y+xy

=> 5y+3xy = 30y+2xy

=> xy= 25y

=> x=25.

Hence the original pair of brown socks = 25.

6)   A two number committee comprising of one male and the female number is to be constituted out of five male and 3 females. Amongst the females, Ms. A refused to be number committee in which Mr.B is taken as the number of how many different ways can the committee be constituted?

a. 11
b. 30
c. 14
d. 20
Answer  Explanation 

ANSWER: 14

Explanation:
For each visualization let us name the females and Females(3) Males(5)

A B C D E F G H

Since A cannot go with B.

She will make team with four males in four ways AD,AF,AG,AH.

Since there is no compulsion with females C and E.

They can make team with 5 males in 5 different ways each.

Therefore, Total number of ways = 4+5+5 =14

7)   A selection is to be made for one post of principal and two posts of vice-principal amongst the six candidates called for the interview only two are eligible for the post of principal while they all are eligible for the post of vice-principal. The number of possible combinations of selectees is___________?

a. 4
b. 12
c. 18
d. None of these
Answer  Explanation 

ANSWER: None of these

Explanation:
Total number of ways = 2C1 . 5C2 = 2 x 5!/3!2! = 2 x 10 = 210

8)   A student has to opt for 2 subjects out of 5 subjects for a course. Namely commerce, economics, statistics, mathematics 1 and Mathematics 2, Mathematics 2 can be offered only if mathematics 1 has also opted. The number of different combinations of two subjects which can be opted is_________?

a. 5
b. 6
c. 7
d. 18
Answer  Explanation 

ANSWER: 7

Explanation:
Number of ways of opting a subject other than Mathematics II = 4C2. = 4x3x2!/2!x2 = 6.

Number of ways of selection of Mathematics II = 1

Therefore, Total Number of ways = 6+1 =7.

9)   2 men and 1 woman board a bus of which 5 seats are vacant, one of these 5 seats is reserved for ladies. A woman may or may not sit on the seat reserved for ladies, In how many different ways can the five seats be occupied by these passengers?

a. 15
b. 36
c. 48
d. 60
Answer  Explanation 

ANSWER: 36

Explanation:
Case I if lady sits on the reserved seat, then 2 men can occupy seats from 4 vacant seats in = 4P2 = 4×3 = 12ways

Case II if lady does not sit on reversed seat, then I. Woman can occupy a seat from four seats in 4 ways. I. man can occupy a seat from 3 seats in 3 ways, also I. man left can occupy a seat from remaining two seats in 2 ways.

Therefore, Total ways = 4x3x2 = 24ways

From case I and case II

Total number of ways = 12+24 = 36

10)   Three flags each of different colours are available for a military exercise, Using these flags different codes can be generated by waving
I. Single flag of different colours
II. Any two flags in a different sequence of colours.
III. three flags in a different sequence of colours.
The maximum number of codes that can be generated is.

a. 6
b. 9
c. 15
d. 18
Answer  Explanation 

ANSWER: 15

Explanation:
This type of question becomes very easy when we assume three colour are red(R) blue(B) and Green(G).

We can choose any colour.

Now according to the statement 1 i.e.., codes can be generated by waving single flag of different colours, then number of ways are three i.e.., R.B.G from statement III three flags in different sequence of colours, then number of ways are six i.e.., RBG, BGR, GBR, RGB, BRG, GRB.

Hence total number of ways by changing flag = 3+ 6 +6 = 15

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