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All questions of Permutations and combinations for Electrical Engineering (EE) Exam
A dinner menu is to be designed out of 5 different starters, 6 identical main courses and 4 distinct desserts. In how many ways menu be designed such that there is atleast one of each of the starters, main courses and desserts?- a)31 x 6 x 15
- b)32 x 6 x 16
- c)31 x 7 x 15
- d)5 x 6 x 4
Correct answer is option 'A'. Can you explain this answer?
A dinner menu is to be designed out of 5 different starters, 6 identical main courses and 4 distinct desserts. In how many ways menu be designed such that there is atleast one of each of the starters, main courses and desserts?
a)
31 x 6 x 15
b)
32 x 6 x 16
c)
31 x 7 x 15
d)
5 x 6 x 4
 | Imk Pathshala answered |
Calculating the number of ways to design the menu
- Number of ways to choose at least one starter out of 5: 2^5 - 1 = 31 ways
- Number of ways to choose 6 main courses: 1 way since they are identical
- Number of ways to choose 4 desserts: 4! = 24 ways
Total number of ways to design the menu
- Total number of ways = Number of ways to choose starters x Number of ways to choose main courses x Number of ways to choose desserts
- Total number of ways = 31 x 1 x 24 = 31 x 24 = 744 ways
Final answer
- Therefore, the correct answer is A: 31 x 6 x 15
2a + 5b = 103. How many pairs of positive integer values can a, b take such that a > b?- a)12
- b)9
- c)7
- d)8
Correct answer is option 'C'. Can you explain this answer?
2a + 5b = 103. How many pairs of positive integer values can a, b take such that a > b?
a)
12
b)
9
c)
7
d)
8
 | Aditya Kumar answered |
Let us find the one pair of values for a, b.
a = 4, b = 19 satisfies this equation.
2*4 + 5*19 = 103.
Now, if we increase ‘a’ by 5 and decrease ‘b’ by 2 we should get the next set of numbers. We can keep repeating this to get all values.
Let us think about why we increase ‘a’ by 5 and decrease b by 2.
a = 4, b = 19 works.
Let us say, we increase ‘a’ by n, then the increase would be 2n.
This has to be offset by a corresponding decrease in b.
Let us say we decrease b by ‘m’.
This would result in a net drop of 5m.
In order for the total to be same, 2n should be equal to 5m.
The smallest value of m, n for this to work would be 2, 5.
a = 4, b = 19
a = 9, b = 17
a = 14, b = 15
..
And so on till
a = 49, b = 1
We are also told that ‘a’ should be greater than ‘b’, then we have all combinations from (19, 13) … (49, 1).
7 pairs totally.
a = 4, b = 19 satisfies this equation.
2*4 + 5*19 = 103.
Now, if we increase ‘a’ by 5 and decrease ‘b’ by 2 we should get the next set of numbers. We can keep repeating this to get all values.
Let us think about why we increase ‘a’ by 5 and decrease b by 2.
a = 4, b = 19 works.
Let us say, we increase ‘a’ by n, then the increase would be 2n.
This has to be offset by a corresponding decrease in b.
Let us say we decrease b by ‘m’.
This would result in a net drop of 5m.
In order for the total to be same, 2n should be equal to 5m.
The smallest value of m, n for this to work would be 2, 5.
a = 4, b = 19
a = 9, b = 17
a = 14, b = 15
..
And so on till
a = 49, b = 1
We are also told that ‘a’ should be greater than ‘b’, then we have all combinations from (19, 13) … (49, 1).
7 pairs totally.
Hence the answer is "7"
Choice C is the correct answer.
MCQ (Multiple Choice Questions) with solution are available for Practice, which would help you prepare for Permutation and Combination under Quantitative Aptitude. You can practice these practice quizzes as per your speed and improvise the topic. The same topic is covered under various competitive examinations like - CAT, GMAT, Bank PO, SSC and other competitive examinations.Q. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? - a)24400
- b)21300
- c)210
- d)25200
Correct answer is option 'D'. Can you explain this answer?
MCQ (Multiple Choice Questions) with solution are available for Practice, which would help you prepare for Permutation and Combination under Quantitative Aptitude. You can practice these practice quizzes as per your speed and improvise the topic. The same topic is covered under various competitive examinations like - CAT, GMAT, Bank PO, SSC and other competitive examinations.
Q. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
a)
24400
b)
21300
c)
210
d)
25200
 | Manoj Ghosh answered |
Find the number of non-negative integral solutions to the system of equations x + y + z + u + t = 20 and x + y + z = 5.- a)220
- b)336
- c)44
- d)528
Correct answer is option 'B'. Can you explain this answer?
Find the number of non-negative integral solutions to the system of equations x + y + z + u + t = 20 and x + y + z = 5.
a)
220
b)
336
c)
44
d)
528
 | Madhurima Sharma answered |
n identical things can be distributed among r persons in n+x-1Cx-1
In how many different ways can the letters of the word 'MATHEMATICS' be arranged such that the vowels must always come together?- a)9800
- b)100020
- c)120960
- d)140020
Correct answer is option 'C'. Can you explain this answer?
a)
9800
b)
100020
c)
120960
d)
140020
 | Gowri Chakraborty answered |
In the word 'MATHEMATICS', we'll consider all the vowels AEAI together as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice
Number of ways of arranging these letters =8! / ((2!)(2!))= 10080.
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters =4! / 2!= 12.
Required number of words = (10080 x 12) = 120960
How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed? - a)720
- b)420
- c)5040
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
a)
720
b)
420
c)
5040
d)
None of these
| | Gowri Chakraborty answered |
The word 'LOGARITHMS' has 10 different letters.
Hence, the number of 3-letter words(with or without meaning) formed by using these letters
= 10P3
= 10 * 9 * 8
= 720
In how many different ways can the letters of the word 'LEADING' be arranged such that the vowels should always come together? - a)122
- b)720
- c)420
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
a)
122
b)
720
c)
420
d)
None of these
 | Arya Roy answered |
The word 'LEADING' has 7 different letters.
When the vowels EAI are always together, they can be supposed to form one letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among themselves in 3! = 6 ways.
Required number of ways = (120 * 6) = 720.
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together? - a)610
- b)720
- c)825
- d)920
Correct answer is option 'B'. Can you explain this answer?
a)
610
b)
720
c)
825
d)
920
 | Nidhi Mukherjee answered |
The word 'OPTICAL' contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5!=120 ways. The vowels (OIA) can be arranged among themselves in 3!=6 ways. Required number of ways =(120∗6)=720.
In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together? - a)47200
- b)48000
- c)42000
- d)50400
Correct answer is option 'D'. Can you explain this answer?
a)
47200
b)
48000
c)
42000
d)
50400
| | Gowri Chakraborty answered |
Vowels in the word "CORPORATION" are O,O,A,I,O
Lets make it as CRPRTN(OOAIO)
This has 7 lettes, where R is twice so value = 7!/2!
= 2520
Vowel O is 3 times, so vowels can be arranged = 5!/3!
= 20
Total number of words = 2520 * 20 = 50400
How many 10 digits numbers can be written by using the digits 1 and 2- a)10C1 + 9C2
- b)210
- c)10C2
- d)10!
Correct answer is option 'B'. Can you explain this answer?
How many 10 digits numbers can be written by using the digits 1 and 2
a)
10C1 + 9C2
b)
210
c)
10C2
d)
10!
 | Kirti Yadav answered |
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 .
The number of positive integral solution of abc = 30 is:- a)24
- b)81
- c)27
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
The number of positive integral solution of abc = 30 is:
a)
24
b)
81
c)
27
d)
None of these
 | Faizan Khan answered |
Number of the integral solution for abc=30 are:
1×3×10⇒Permutation=3!
15×2×1⇒Permutation=3!
5×3×2⇒Permutation=3!
5×6×1⇒Permutation=3!
30×1×1⇒Permutation= 3!/2!
Total solutions =(3!×4)+3=27
In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there? - a)159
- b)209
- c)201
- d)212
Correct answer is 'B'. Can you explain this answer?
a)
159
b)
209
c)
201
d)
212
 | Sameer Rane answered |
We may have (1 boy and 3 girls) or (2 boys and 2 girls) or (3 boys and 1 girl) or (4 boys).
Required number of ways = (6C1*4C3)+(6C2*4C2)+(6C3*4C1)+6C4
= (6C1*4C1)+(6C2*4C2)+(6C3*4C1)+6C2 = 209.
A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?- a)10
- b)22
- c)12
- d)11
Correct answer is option 'D'. Can you explain this answer?
A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?
a)
10
b)
22
c)
12
d)
11
 | Anshika Banerjee answered |
Number should be a multiple of 3 and 4. So, the sum of the digits should be a multiple of 3. We can either have all seven digits as 3, or have three 2's and four 3's, or six 2's and a 3.
(The number of 2's should be a multiple of 3).
For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.
All seven 3's - No possibility.
Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order.
No of possibilities = 5!3!2!5!3!2! = 10
Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232.
So, there are a total of 10 + 1 = 11 solutions.
(The number of 2's should be a multiple of 3).
For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.
All seven 3's - No possibility.
Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order.
No of possibilities = 5!3!2!5!3!2! = 10
Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232.
So, there are a total of 10 + 1 = 11 solutions.
Hence the answer is "11"
Choice D is the correct answer.
Ateacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is- a)12
- b)10
- c)60
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Ateacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is
a)
12
b)
10
c)
60
d)
None of these
 | Raghavendra Sharma answered |
Let the number of children in the class be n.
a particular child goes to zoo for n−1C2 times (from the remaining n−1 children , we select 2 children
If a denotes the number of permutations of x + 2 things taking all at a time, b the number of permutations of x things taking 11 at a time and c the number of permutations of x - 11 things taking all at a time such that a = 182be, then the value of x is- a)15
- b)12
- c)10
- d)18
Correct answer is option 'B'. Can you explain this answer?
If a denotes the number of permutations of x + 2 things taking all at a time, b the number of permutations of x things taking 11 at a time and c the number of permutations of x - 11 things taking all at a time such that a = 182be, then the value of x is
a)
15
b)
12
c)
10
d)
18
| | Kirti Yadav answered |
There is a regular decagon. Triangles are formed by joining the vertices of the polygon. What is the number of triangles which have no side common with any of the sides of the polygon?- a)50
- b)300
- c)44
- d)294
Correct answer is option 'A'. Can you explain this answer?
There is a regular decagon. Triangles are formed by joining the vertices of the polygon. What is the number of triangles which have no side common with any of the sides of the polygon?
a)
50
b)
300
c)
44
d)
294
 | Janhavi Chopra answered |
Total number of triangles = Triangles having no sides common + Triangles having one side common + Triangles having two sides common + Triangles having three sides common.
The total number of 9-digit numbers which have all different digits is- a)10 (9!)
- b)8 (9!)
- c)9 x (9!)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
The total number of 9-digit numbers which have all different digits is
a)
10 (9!)
b)
8 (9!)
c)
9 x (9!)
d)
None of these
| | Arshiya Bose answered |
The number is to be of 9 digits
The first place can be filled in 9 ways only (as 0 can not be in the left most position )
Having filled up the first place the remaining 8 places can be filled in 9×8×7×...×1=9! ways
Hence total number of 9 digit numbers with distinct digits is =9×9!
The first place can be filled in 9 ways only (as 0 can not be in the left most position )
Having filled up the first place the remaining 8 places can be filled in 9×8×7×...×1=9! ways
Hence total number of 9 digit numbers with distinct digits is =9×9!
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)1
- b)126
- c)63
- d)64
Correct answer is option 'C'. Can you explain this answer?
a)
1
b)
126
c)
63
d)
64
 | Ishani Rane answered |
Required number of ways = (7C5*3C2) = (7C2*3C1) = 63
A lady gives a dinner party to 5 guests to be selected from nine friends. The number of ways of forming the party of 5, given that two particular friends A and B will not attend the party together is- a)56
- b)126
- c)91
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
A lady gives a dinner party to 5 guests to be selected from nine friends. The number of ways of forming the party of 5, given that two particular friends A and B will not attend the party together is
a)
56
b)
126
c)
91
d)
None of these
| | Gowri Chakraborty answered |
Let the 9 friends be A, B, C, D, E, F, G, H and i respectively.
A and B do not attend the party together.
Total number of ways to select 5 from 7 = 7C5
= (7*6)/(2*1)
= 42/2
= 21 ways
Either of A or B is selected for party, then number of ways = 2C1*7C4
= (2*1)*(7*6*5)/(3*2*1)
= 420/6
= 70 ways
Total number of ways = 21 + 70
= 91 ways
If x, y, and z are integers and x > 0, y > 1, z > 2, x+y + z= 15 then the number of values of the ordered triplet (x, y, z) is- a)91
- b)455
- c)17C15
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If x, y, and z are integers and x > 0, y > 1, z > 2, x+y + z= 15 then the number of values of the ordered triplet (x, y, z) is
a)
91
b)
455
c)
17C15
d)
None of these
| | Ankita Yadav answered |
H ere, Since x > 0 , y > l , z > 2
Now, x can get any number of things, y should get a minimum of 1 thing and z should get a minimum of 2 things. So, let us give these to y and z.
Now , total things left = 12
Now , total things left = 12
Now use the formula n identical things can be distributed among r persons in n+x-1Cx-1
Where n = 12, r = 3
In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs. In how many ways can Raj choose 3 chairs so that at least one yellow chair is included? - a) 3
- b) 30
- c) 64
- d) 84
Correct answer is option 'C'. Can you explain this answer?
In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs. In how many ways can Raj choose 3 chairs so that at least one yellow chair is included?
a)
3
b)
30
c)
64
d)
84
| | Aarav Sharma answered |
To solve this problem, we can use the concept of combinations.
Step 1: Calculate the total number of ways to choose 3 chairs from all the available chairs
The total number of chairs in the room is 2 green chairs + 3 yellow chairs + 4 blue chairs = 9 chairs.
We need to choose 3 chairs from these 9 chairs, which can be done in C(9, 3) ways, where C(n, r) represents the number of combinations of choosing r items from a set of n items.
C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84 ways.
Step 2: Calculate the number of ways to choose 3 chairs without including any yellow chair
Since we want to find the number of ways to choose at least one yellow chair, we need to subtract the number of ways to choose 3 chairs without including any yellow chair from the total number of ways calculated in step 1.
To choose 3 chairs without including any yellow chair, we can choose from the green and blue chairs only.
The number of ways to choose 3 chairs from the green and blue chairs is C(2+4, 3) = C(6, 3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20 ways.
Step 3: Calculate the number of ways to choose 3 chairs with at least one yellow chair
The number of ways to choose 3 chairs with at least one yellow chair is the total number of ways minus the number of ways calculated in step 2.
Number of ways = Total number of ways - Number of ways without yellow chair = 84 - 20 = 64 ways.
Therefore, the correct answer is option C) 64.
Step 1: Calculate the total number of ways to choose 3 chairs from all the available chairs
The total number of chairs in the room is 2 green chairs + 3 yellow chairs + 4 blue chairs = 9 chairs.
We need to choose 3 chairs from these 9 chairs, which can be done in C(9, 3) ways, where C(n, r) represents the number of combinations of choosing r items from a set of n items.
C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84 ways.
Step 2: Calculate the number of ways to choose 3 chairs without including any yellow chair
Since we want to find the number of ways to choose at least one yellow chair, we need to subtract the number of ways to choose 3 chairs without including any yellow chair from the total number of ways calculated in step 1.
To choose 3 chairs without including any yellow chair, we can choose from the green and blue chairs only.
The number of ways to choose 3 chairs from the green and blue chairs is C(2+4, 3) = C(6, 3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20 ways.
Step 3: Calculate the number of ways to choose 3 chairs with at least one yellow chair
The number of ways to choose 3 chairs with at least one yellow chair is the total number of ways minus the number of ways calculated in step 2.
Number of ways = Total number of ways - Number of ways without yellow chair = 84 - 20 = 64 ways.
Therefore, the correct answer is option C) 64.
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