Recent questions and answers in Combinatory

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1

Applied Test Series
A Professor tells 3 Jokes in his maths class each year. How large a set of jokes does the professor need in order never to repeat the exact same triple of jokes over a period of 12 years?_________

Sudhanshu10 answered in Combinatory 5 days ago

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32 votes

5 answers

2

GATE CSE 2003 | Question: 5
$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)

Arpit Patel answered in Combinatory Oct 13

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51 votes

14 answers

3

GATE CSE 2015 Set 3 | Question: 5
The number of $4$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $\{1, 2, 3\}$ is ________.

Kanwae Kan answered in Combinatory Oct 11

8.8k views

1 vote

1 answer

4

gateacademy test series
There are 12 stations on a rail route. How many ways a special train can stop at 4 of these stations, so that no two stops are consecutive stations? please explain in detail

Nibchee answered in Combinatory Sep 16

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1 answer

5

gateacademy testseries
In how many ways can seven different jobs be assigned to four different employees so that each employee is assigned at least one job and the most difficult job is assigned to the best employee?

nishant.singhal.cs answered in Combinatory Sep 8

by nishant.singhal.cs

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1 answer

6

Kenneth H Rosen 7th edition
Please see example 6. l am not getting the mathematical insight. Can anyone please tell how they are arriving at the answer.

Sahiltam answered in Combinatory Sep 7

292 views

3 votes

3 answers

7

KENETH ROSEN
How many different ways are there to seat four people around a circular table, where two seatings are considered the same when each person has the same left neighbor and the same right neighbor? ANSWER IS 6 OR 3 .????

M_eight answered in Combinatory Sep 7

1.3k views

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1 answer

8

Kenneth Rosen Edition 7th Exercise 6.4 Question 16 (Page No. 421)
Use question $14$ and Corollary $1$ to show that if $n$ is an integer greater than $1,$ then $\binom{n}{\left \lfloor n/2 \right \rfloor}\geq \frac{2^{n}}{2}.$ Conclude from part $(A)$ that if $n$ is a positive integer, then $\binom{2n}{n}\geq \frac{4^{n}}{2n}.$

harsh_sojitra answered in Combinatory Sep 6

by harsh_sojitra

51 views

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3 answers

9

Kenneth Rosen Edition 7th Exercise 6.1 Question 2 (Page No. 396)
An office building contains $27$ floors and has $37$ offices on each floor. How many offices are in the building?

harsh_sojitra answered in Combinatory Sep 6

by harsh_sojitra

78 views

19 votes

4 answers

10

CMI2014-A-01
For the inter-hostel six-a-side football tournament, a team of $6$ players is to be chosen from $11$ players consisting of $5$ forwards, $4$ defenders and $2$ goalkeepers. The team must include at least $2$ forwards, at least $2$ defenders and at least $1$ goalkeeper. Find the number of different ways in which the team can be chosen. $260$ $340$ $720$ $440$

Nirmalya Pratap 1 answered in Combinatory Sep 5

by Nirmalya Pratap 1

1.4k views

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1 answer

11

Kenneth Rosen Edition 7th Exercise 6.4 Question 18 (Page No. 421)
Suppose that $b$ is an integer with $b \geq 7.$ Use the binomial theorem and the appropriate row of Pascal’s triangle to find the base-$b$ expansion of $(11)^{4}_{b}$ [that is, the fourth power of the number $(11)_{b}$ in base-$b$ notation].

harsh_sojitra answered in Combinatory Aug 23

by harsh_sojitra

83 views

48 votes

16 answers

12

GATE CSE 2016 Set 1 | Question: 26
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.

Nagasaikanmatha answered in Combinatory Aug 1

by Nagasaikanmatha

16.7k views

52 votes

9 answers

13

GATE CSE 2004 | Question: 75
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$

Arkaprava answered in Combinatory Jul 26

8.7k views

15 votes

7 answers

14

GATE CSE 2008 | Question: 24
Let $P =\sum \limits_ {i\;\text{odd}}^{1\le i \le 2k} i$ and $Q = \sum\limits_{i\;\text{even}}^{1 \le i \le 2k} i$, where $k$ is a positive integer. Then $P = Q - k$ $P = Q + k$ $P = Q$ $P = Q + 2k$

ShivangiChauhan answered in Combinatory Jul 21

by ShivangiChauhan

2.5k views

10 votes

2 answers

15

GATE CSE 1990 | Question: 3-ix
The number of ways in which $5\; A's, 5\; B's$ and $5\; C's$ can be arranged in a row is: $15!/(5!)^{3}$ $15!$ $\left(\frac{15}{5}\right)$ $15!(5!3!)$.

subbus answered in Combinatory Jul 17

by subbus

1.7k views

0 votes

3 answers

16

ISI2019-MMA-20
Suppose that the number plate of a vehicle contains two vowels followed by four digits. However, to avoid confusion, the letter $‘O’$ and the digit $‘0’$ are not used in the same number plate. How many such number plates can be formed? $164025$ $190951$ $194976$ $219049$

NastyBall answered in Combinatory Jul 9

1.7k views

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1 answer

17

ISI2017-MMA-6
In a class of $80$ students, $40$ are girls and $40$ are boys. Also, exactly $50$ students wear glasses. Then the set of all possible numbers of boys without glasses is $\{0, \dots , 30\}$ $\{10, \dots , 30\}$ $\{0, \dots , 40\}$ none of these

pranabdbg14 answered in Combinatory Jul 3

398 views

4 votes

3 answers

18

CMI2015-A-10
The school athletics coach has to choose $4$ students for the relay team. He calculates that there are $3876$ ways of choosing the team if the order in which the runners are placed is not considered. How many ways are there of choosing the team if the order of the runners is to be ... $12,000$ and $25,000$ Between $75,000$ and $99,999$ Between $30,000$ and $60,000$ More than $100,000$

soujanyareddy13 answered in Combinatory May 6

by soujanyareddy13

325 views

2 votes

2 answers

19

CMI2015-B-02
Consider a social network with $n$ persons. Two persons $A$ and $B$ are said to be connected if either they are friends or they are related through a sequence of friends: that is, there exists a set of persons $F_1, \dots, F_m$ such that $A$ and ... . It is known that there are $k$ persons such that no pair among them is connected. What is the maximum number of friendships possible?

soujanyareddy13 answered in Combinatory May 6

by soujanyareddy13

309 views

2 votes

3 answers

20

CMI2017-B-4
In a party there are $2n$ participants, where $n$ is a positive integer. Some participants shake hands with other participants. It is known that there are no three participants who have shaken hands with each other. Prove that the total number of handshakes is not more than $n^2.$

soujanyareddy13 answered in Combinatory May 6

by soujanyareddy13

480 views

28 votes

4 answers

21

TIFR2015-A-8
There is a set of $2n$ people: $n$ male and $n$ female. A good party is one with equal number of males and females (including the one where none are invited). The total number of good parties is. $2^{n}$ $n^{2}$ $\binom{n}{⌊n/2⌋}^{2}$ $\binom{2n}{n}$ None of the above

mayankso answered in Combinatory May 4

2.4k views

1 vote

2 answers

22

CMI2018-A-5
How many paths are there in the plane from $(0,0)$ to $(m,n)\in \mathbb{N}\times \mathbb{N},$ if the possible steps from $(i,j)$ are either $(i+1,j)$ or $(i,j+1)?$ $\binom{2m}{n}$ $\binom{m}{n}$ $\binom{m+n}{n}$ $m^{n}$

soujanyareddy13 answered in Combinatory May 4

by soujanyareddy13

222 views

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1 answer

23

CMI2019-B-3
There is a party of $n$ people. Each attendee has at most $r$ friends in the party. The friend circle of a person includes the person and all her friends. You are required to pick some people for a party game, with the restriction that at most one person is picked from each friend circle. Show that you can pick $\dfrac{n}{r^{2}+1}$ people for the game.

soujanyareddy13 answered in Combinatory May 4

by soujanyareddy13

306 views

3 votes

3 answers

24

CMI2019-A-5
Let $\pi=[x_{1},x_{2},\cdots,x_{n}]$ be a permutation of $\{1,2,\cdots,n\}.$ For $k<n,$ we say that $\pi$ has its first ascent at $k$ if $x_{1}>x_{2}\cdots>x_{k}$ and $x_{k}<x_{k+1}.$ How many permutations have their first ascent at $k?$ $\binom{n}{k}-\binom{n}{(k+1)}$ $\frac{n!}{k!}-\frac{n!}{(k+1)!}$ $\frac{n!}{(k+1)!}-\frac{n!}{(k+2)!}$ $\binom{n}{(k+1)}-\binom{n}{(k+2)}$

soujanyareddy13 answered in Combinatory May 4

by soujanyareddy13

287 views

2 votes

2 answers

25

CMI2019-A-4
There are five buckets, each of which can be open or closed. An arrangement is a specification of which buckets are open and which bucket are closed. Every person likes some of the arrangements and dislikes the rest. You host a party, and amazingly, no two people on the guest list have the ... What is the maximum number of guests possible? $5^{2}$ $\binom{5}{2}$ $2^{5}$ $2^{2^{5}}$

soujanyareddy13 answered in Combinatory May 4

by soujanyareddy13

289 views

7 votes

2 answers

26

GATE CSE 2021 Set 1 | Question: 19
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that: The fastest computer gets the toughest job and the slowest computer gets the easiest job. Every computer gets at least one job. The number of ways in which this can be done is ___________.

varunrajarathnam answered in Combinatory Apr 27

by varunrajarathnam

2.3k views

7 votes

5 answers

27

GATE CSE 2021 Set 2 | Question: 50
Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________

Debdeep1998 answered in Combinatory Apr 9

2.5k views

1 vote

3 answers

28

P and C doubt
Given a 4 *4 grid points , how many Triangles with vertices on the grid can be formed?

reboot answered in Combinatory Feb 24

by reboot

429 views

1 vote

0 answers

29

JEST2020
For two n-bit strings x, y ∈ {0, 1}n, define z := x ⊕ y to be the bitwise XOR of the two strings (that is, if xi, yi, zi denote the i-th bits of x, y, z respectively, then zi = xi + yi mod 2). A function h : {0, 1}n → {0, 1}n is called linear if h(x ⊕ y) = h(x) ⊕ h(y), for every x, y ∈ {0, 1}n. The number of such linear functions for n ≥ 2: 2^n 2^2n 2^(n+1) n

vivek_mishra asked in Combinatory Feb 15

by vivek_mishra

238 views

13 votes

14 answers

30

GATE CSE 2019 | Question: 21
The value of $3^{51} \text{ mod } 5$ is _____

SlashSurfer answered in Combinatory Feb 2

11.9k views

2 votes

2 answers

31

UGCNET-Oct2020-II: 2
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books? $4$ $6$ $7$ $9$

jothee asked in Combinatory Nov 20, 2020

by jothee

1.3k views

2 votes

1 answer

32

K H Rosen Ex
15. a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and ... cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?

Sanjay Sharma asked in Combinatory Jul 4, 2020

by Sanjay Sharma

412 views

4 votes

1 answer

33

Counting number of pairs whose sum is less than k
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.

dd asked in Combinatory Jun 8, 2020

by dd

602 views

0 votes

1 answer

34

Kenneth Rosen Edition 7th Exercise 8.3 Question 16 (Page No. 535)
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

577 views

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1 answer

35

Kenneth Rosen Edition 7th Exercise 8.3 Question 15 (Page No. 535)
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

224 views

0 votes

2 answers

36

Kenneth Rosen Edition 7th Exercise 8.3 Question 14 (Page No. 535)
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

454 views

0 votes

1 answer

37

Kenneth Rosen Edition 7th Exercise 8.3 Question 13 (Page No. 535)
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

269 views

1 vote

2 answers

38

Kenneth Rosen Edition 7th Exercise 8.3 Question 12 (Page No. 535)
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

220 views

0 votes

1 answer

39

Kenneth Rosen Edition 7th Exercise 8.3 Question 11 (Page No. 535)
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

153 views

0 votes

1 answer

40

Kenneth Rosen Edition 7th Exercise 8.3 Question 10 (Page No. 535)
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$

Lakshman Patel RJIT asked in Combinatory May 10, 2020

by Lakshman Patel RJIT

122 views

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