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Test: Basic Concepts Of Permutations And Combinations- 3 - CA Foundation MCQ


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30 Questions MCQ Test Quantitative Aptitude for CA Foundation - Test: Basic Concepts Of Permutations And Combinations- 3

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Test: Basic Concepts Of Permutations And Combinations- 3 - Question 1

 A man has 3 sons and 6 schools within his reach. In how many ways, he can send them to school, if no two of his sons are to read in the same school?

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 2

 If 13C6 + 2 13C5 + 13C4 = 15Cx then, x = _________.

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Test: Basic Concepts Of Permutations And Combinations- 3 - Question 3

If six times the number of permutations of n items taken 3 at a time is equal to seven times the number of permutation of (n-1) items taken 3 at a time, then the value of n will be: 

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 4

The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is:

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 5

 If 1000C98 = 999C97 + xc901 , then the value of x will be:

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 6

Number of ways of shaking hands in a group of 10 persons shaking hands to each other are:

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 7

 If 15C3r = 15Cr+3, then r is equal is 

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 8

 If six times the number of permutations of n items taken 3 at a time is equal to seven times the number of permutation of (n-1) items taken 3 at a time, then the value of n will be: 

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 9

The ways of selecting 4 letters from the word EXAMINATION is 

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 10

The letters of the word "VIOLENT" are arranged so that the vowels occupy even place only. The number of permutations is ______.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 11

 How many permutations can be formed from the letters of the word  " DRAUGHT" if both vowels may not be separated?

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 11

The vowels are never being separated.

So, vowels a & u are taken together and considered as one letter.

So, there are 6 letters in the word draught

The number of ways in which it can be arranged are given by =6!=720. 

But the internal arrangement can be done of a and u in 2 ways.

Hence the total arrangements are 720×2=1440.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 12

 If nP4 = 20 (nP2) then the value of n is ______________.

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 12

nP4 represents the number of permutations of n things taken 4 at a time, and nP2 represents the number of permutations of n things taken 2 at a time.

We are given that nP4 = 20 (nP2).

Using the formula for permutations, we know that nP4 = n!/(n-4)!, and nP2 = n!/(n-2)!.

Substituting these expressions into the given equation, we get:

n!/(n-4)! = 20 * n!/(n-2)!

Cancelling out the common factor of n!/[(n-4)! * (n-2)!], we get:

(n-3) * (n-2) = 20

Expanding the left side of the equation, we get:

n2 - 5n + 6 = 20

Simplifying, we get:

n2 - 5n - 14 = 0

Factorizing, we get:

(n - 7) * (n + 2) = 0

Therefore, the solutions are n = 7 and n = -2.

However, n cannot be negative since we are counting permutations. Therefore, the only possible value of n is 7.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 13

A polygon has 44 diagonals then the number of its sides are:

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 14

 How many different words can be formed with the letters of the word "LIBERTY"

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 15

You have to make a choice of 4 balls out of one red one blue and ten white balls. The numbers of  ways in which  this can be done to include the red ball  but exclude the blue ball always is ________.

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 15

To include the red ball and exclude the blue ball, we need to choose 3 remaining balls from the 10 white balls. This can be done in 10C3 ways.

10C3 = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = 10*9*8 / (3*2*1) = 120

So, the answer is 10C3 or 120 ways. The correct option is:

10C3

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 16

You have to make a choice of 4 balls out of one red one blue and ten white balls. The numbers of ways this can be done to always include the red ball is _______.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 17

In a cross word puzzle 20 words are to be guessed of which 8 words have each an alternative solution. The number of possible solution is ________.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 18

How many combinations can be formed of eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination?

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 18

We have,1,2,3,4,5,6,7,8

Total counters = 8

Taken at a time = 4

The total combinations without restrictions= 8C4 = 70

Odd = 1,3,5,7

Even = 2,4,6,8

We have to find out those combinations which contain at least one even and one odd counter i.e. those neither can have all 4 even counters nor all 4 odd counters.

Let's first find the the no. of combinations with restrictions i.e. which contains all even or all odd counters.

So, the no. of combination which have all 4 odd counters = 4C4 = 1

Similarly, the no. of combination which have all 4 even counters = 4C4 = 1

Hence, the required no. combinations which contain at least one even and one odd counters = Total - Restricted combinations

= 70–1–1

= 68 Ans.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 19

Find the number of ways in which a selection of four letters can be made from the letters of the word 'PROPORTION,

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 19

Given the word is

PROPORTION

When

Number of P is =2

Number of O is =3

Number of R is =2

Number of I is =1

Number of T is =1

Number of N is =1

Let us frist find the number of selections.

Case 1:- All four letters distinct = number of ways will be 4×6C4​=4!×4!(6−4)!6!​=2!6×5×4×3×2×1​=360

Case 2:- Three letters same and one letter distinct, then no. of ways  = 3!4!​5C1​=3!4×3!​×5=20

Case 3:- Two letters of one type and the other two letters of other type, then no. of ways(out for P,R and O)=3C2​×4C2​=3×2!×2!4!​=18

Case 4:-two letters same and other two letters are different.then, no. of ways. = 3C1​×5C2​=3×10=30×2!4!​=360

Now according to given question,

Total number of selections=15+5+3+30=53 

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 20

Out of 6 members belonging to party A and 4 to party B in how many ways a committee of 5 can be selected so that members of party A are in majority?

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 21

A team of 5 is to be selected from 8 boys and three girls. Find the probability that it includes two particular girls.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 22

You have to make a choice of 4 balls out of one red one blue and ten white balls. The number of ways in which this can be done to exclude both the red and blues ball is ________.

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 22

Total no. of balls =12

They said to exclude both red&blue
then, remaining total balls=10
out of this 4 should be selected
therefore, Req. answer=10c4

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 23

The number of words which can be formed with 2 different consonants and 1 vowel out of 7 different consonants and 3 different vowels the vowel to lie between 2 consonants is _________.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 24

A question paper divided into two groups consisting of 3 and 4 questions respectively carries the note “It is not necessary to answer all the questions. One question must be answered from each group”. In how many ways you can select these questions?

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 24

According to formula:

total no of ways to select questions|
= (3-1)! (4-1)!
= 2! 3!
= 2 x 3 x 2
= 12 ............. ans..

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 25

The number of ways in which four letters of the word MATHEMATICS can be arranged is given by:

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 25

The word contain 11 letters with repetition of letters.
We can choose 4 letters by
i) all the four letters are distinct
ii) Two distinct and two alike
iii) Two alike of one kind and two alike of other kind
i) 8 different letters are M, A, T, H, E, I, C, S
This can be chosen by 8C4 × 4! = 1680
ii) Three pairs alike letters MM, AA, TT. One pair can be chosen by 3C1 = 3 ways 
Remaining two letters can be chosen by 7C
Each such groups has 4 letters out of which 2 are alike.
Hence, it can be chosen by 4!/2! 
Total number of ways = 3 × 7C2 × 4!/2! = 756
iii) Three pairs of 2 alike letters. Two pairs can be chosen by 3C2
 3C2 groups of 4 letters each.
There are 4 letters of which 2 alike of one kind and two alike of other kind in each group.
That can be arranged by 4!/2! × 1/2!
Total number of ways = 3C2 × 4!/2! × 1/2! = 18
Total number of ways = 1680 + 756 + 18 = 2454

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 26

 Find  the number of arrangements of 5 things taken out of 12 things, in which one particular thing must always be included.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 27

 In how many ways 3 prizes out of 5 can be distributed amongst 3 brothers Equally?

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 28

In forming a committee of 5 out of 5 males and 6 females how many choices you have to made so that there are 3 males and 2 females?

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 29

 In how many ways 21 red balls and 19 blue balls can be arranged in a row so that no two blue balls are together?

Detailed Solution for Test: Basic Concepts Of Permutations And Combinations- 3 - Question 29

As there are 21 red balls and 19 blue balls.

So there can be 22 different positions for blue balls ( because we can't arrange two blue balls together).

Applying concept,

Total ways = 22C19

= 22!/19!(22-19)!

= 22!/19!*3!

= 22*21*20*19!/19!*3!

= 22*7*10

= 1540.

Test: Basic Concepts Of Permutations And Combinations- 3 - Question 30

In forming a committee of 5 out of 5 males and 6 females how many choices you have to made so that there are 3 males and 2 females? How many choices you have to make if there are 2 males?

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