Recent activity in Combinatory / GATE Overflow for GATE CSE

3 votes

2 answers

1

GO Classes 2023 | IIITH Mock Test 1 | Question: 4
How many ways are there to arrange the $12$ letters of $\text{AAABBBBCCCCC}$ without having two $\text{Cs}$ together? $2652$ $1960$ $1826$ $2260$

How many ways are there to arrange the $12$ letters of $\text{AAABBBBCCCCC}$ without having two $\text{Cs}$ together?$2652$$1960$$1826$$2260$

viral8702

792 views

viral8702 commented 7 hours ago

67 votes

10 answers

2

GATE CSE 2016 Set 1 | Question: 27
Consider the recurrence relation $a_1 =8 , a_n =6n^2 +2n+a_{n-1}$. Let $a_{99}=K\times 10^4$. The value of $K$ is __________.

Consider the recurrence relation $a_1 =8 , a_n =6n^2 +2n+a_{n-1}$. Let $a_{99}=K\times 10^4$. The value of $K$ is __________.

halfcodeblood

29.3k views

halfcodeblood commented 5 days ago

45 votes

4 answers

3

GATE CSE 2016 Set 1 | Question: 2
Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$

Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$?$a_n = a_{n-1}+ 2a_{...

halfcodeblood

9.5k views

halfcodeblood commented 6 days ago

19 votes

18 answers

4

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

The value of $3^{51} \text{ mod } 5$ is _____

halfcodeblood

18.2k views

halfcodeblood commented 6 days ago

38 votes

3 answers

5

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 ___________.

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 g...

halfcodeblood

11.8k views

halfcodeblood commented Apr 11

34 votes

8 answers

6

GATE CSE 2002 | Question: 13
In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, the number of ways is $2$, i.e., $1+2, 2+1$. Give only ... $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.

In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, th...

halfcodeblood

7.3k views

halfcodeblood commented Apr 11

65 votes

16 answers

7

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 ________.

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 ___...

halfcodeblood

15.6k views

halfcodeblood commented Apr 10

38 votes

7 answers

8

GATE CSE 1999 | Question: 2.2
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above

Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves?$1638$$2100$$2640$None of th...

halfcodeblood

12.3k views

halfcodeblood commented Apr 10

37 votes

5 answers

9

GATE CSE 1989 | Question: 4-i
How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.

How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.

halfcodeblood

7.0k views

halfcodeblood commented Apr 10

36 votes

5 answers

10

GATE CSE 2007 | Question: 85
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move to either $(i + 1, j)$ or $(i,j + 1)$. Suppose that the robot is not allowed to traverse the ... $^{20}\mathrm{C}_{10} - ^{8}\mathrm{C}_{4}\times ^{11}\mathrm{C}_{5}$

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move...

halfcodeblood

9.5k views

halfcodeblood commented Apr 10

65 votes

9 answers

11

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? $9$ $8$ $7$ $6$

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 pa...

halfcodeblood

16.7k views

halfcodeblood commented Apr 10

12 votes

2 answers

12

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: $15!/(5!)^{3}$ $15!$ $\left(\frac{15}{5}\right)$ $15!(5!3!)$.

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!)$.

halfcodeblood

3.1k views

halfcodeblood commented Apr 10

38 votes

3 answers

13

GATE IT 2008 | Question: 25
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes? $\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$ $\frac{(n+(b+r)-1)!}{(n-1)!\,(n-1)!\,(b+r)!}$ $\frac{n!}{b!\,r!}$ $\frac{(n + (b + r) - 1)!} {n!\,(b + r - 1)}$

In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes?$\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$$\frac{(n+(b+r)-1)!}{(n-1)!\...

halfcodeblood

8.4k views

halfcodeblood commented Apr 10

0 votes

1 answer

14

Kenneth Rosen Edition 7 Exercise 6.1 Question 32 (Page No. 397)
How many strings of eight uppercase English letters are there if letters can be repeated? if no letter can be repeated? that start with $X,$ if letters can be repeated? that start with $X,$ if no letter can be repeated? that ... can be repeated? that start or end with the letters $BO$ (in that order), if letters can be repeated?

How many strings of eight uppercase English letters are thereif letters can be repeated?if no letter can be repeated?that start with $X,$ if letters can be repeated?that ...

aswinik

596 views

aswinik answered Apr 8

26 votes

5 answers

15

GATE CSE 1994 | Question: 1.15
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$

The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is$n$$n^2$$\frac{n(n-1)}{2}$$\frac{n(n+1)}{2}$

Ice_Cold_V

10.1k views

Ice_Cold_V commented Apr 3

14 votes

3 answers

16

GATE CSE 2023 | Question: 38
Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|=|B|=k$ and $A \cap B=\emptyset$. We say that a permutation of $U$ separates $A$ from $B$ if ... $2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k !)^{2}$

Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|...

ranadip_01

6.6k views

ranadip_01 commented Mar 27

23 votes

9 answers

17

GATE CSE 2003 | Question: 34
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$

$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be place...

꧁༒☬ĿọŗԀ 🆂🅷🅸🆅🅰☬༒꧂

11.2k views

꧁༒☬ĿọŗԀ 🆂🅷🅸🆅🅰☬༒꧂ commented Mar 8

60 votes

9 answers

18

GATE CSE 2005 | Question: 44
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such that, $a \equiv c\mod 3$ and $b \equiv d \mod 5$ $4$ $6$ $16$ $24$

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such th...

Rana-G

13.5k views

Rana-G commented Mar 7

22 votes

1 answer

19

Recurrence Relation - Self Doubt
What is the recurrence relation for the ternary strings of length $n$ which can be constructed using 0,1 or 2 only such that the number of 0’s and number of 1's is odd ?

What is the recurrence relation for the ternary strings of length $n$ which can be constructed using 0,1 or 2 only such that the number of 0’s and number of 1's is od...

SankhaneelM

1.6k views

SankhaneelM commented Mar 6

0 votes

2 answers

20

Kenneth Rosen Edition 7 Exercise 6.1 Question 57 (Page No. 398)
The name of a variable in the JAVA programming language is a string of between $1$ and $65,535$ characters, inclusive, where each character can be an uppercase or a lowercase letter, a dollar sign, an underscore, or a digit, except that the first character must not be a digit. Determine the number of different variable names in JAVA.

The name of a variable in the JAVA programming language is a string of between $1$ and $65,535$ characters, inclusive, where each character can be an uppercase or a lower...

Shriram BM

1.2k views

Shriram BM answered Mar 3

7 votes

4 answers

21

GATE CSE 2023 | Question: 5
The Lucas sequence $L_{n}$ is defined by the recurrence relation: \[ L_{n}=L_{n-1}+L_{n-2}, \quad \text { for } \quad n \geq 3, \] with $L_{1}=1$ and $L_{2}=3$ ... $L_{n}=\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}$

The Lucas sequence $L_{n}$ is defined by the recurrence relation:\[L_{n}=L_{n-1}+L_{n-2}, \quad \text { for } \quad n \geq 3,\]with $L_{1}=1$ and $L_{2}=3$.Which one of t...

Priyam Garg

8.0k views

Priyam Garg answered Feb 27

0 votes

1 answer

22

Kenneth Rosen Edition 7 Exercise 6.5 Question 9 (Page No. 432)
A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are there to choose six bagels? a dozen bagels? two dozen bagels? ... least one of each kind? a dozen bagels with at least three egg bagels and no more than two salty bagels?

A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are th...

naruto.720

7.5k views

naruto.720 commented Feb 24

3 votes

0 answers

23

TIFR CSE 2023 | Part A | Question: 12
Suppose is a piece on a chess board that attacks squares that are exactly two steps in the vertical direction, and squares that are adjacent horizontally (as marked with a "*" figure in the image below) What is the maximum number of these pieces that can ... other? (Note that each square is allowed to contain at most one piece.) $4$ $8$ $16$ $24$ $32$

Suppose is a piece on a chess board that attacks squares that are exactly two steps in the vertical direction, and squares that are adjacent horizontally (as marked with...

RohitParida

500 views

RohitParida commented Feb 21

2 votes

4 answers

24

ACE Test Series: Generating Function
The generating function of the sequence $\left \{ a_{0},a_{1},a_{2}..........a_{n}………...\infty \right \}$ where $a_{n}=\left ( n+2 \right )\left ( n+1 \right ).3^{n}$ is $a)3\left ( 1+3x \right )^{-2}$ $b)3\left ( 1-3x \right )^{-2}$ $c)2\left ( 1+3x \right )^{-3}$ $d)2\left ( 1-3x \right )^{-3}$

The generating function of the sequence $\left \{ a_{0},a_{1},a_{2}..........a_{n}………...\infty \right \}$where $a_{n}=\left ( n+2 \right )\left ( n+1 \right ).3^{n}...

Priyam Garg

1.6k views

Priyam Garg answered Feb 20

24 votes

6 answers

25

TIFR CSE 2012 | Part A | Question: 7
It is required to divide the $2n$ members of a club into $n$ disjoint teams of $2$ members each. The teams are not labelled. The number of ways in which this can be done is: $\frac{\left ( 2n \right )!}{2^{n}}$ $\frac{\left ( 2n \right )!}{n!}$ $\frac{\left ( 2n \right )!}{2^n . n!}$ $\frac{n!}{2}$ None of the above

It is required to divide the $2n$ members of a club into $n$ disjoint teams of $2$ members each. The teams are not labelled. The number of ways in which this can be done ...

Priyam Garg

4.7k views

Priyam Garg answered Feb 18

0 votes

1 answer

26

Kenneth Rosen Edition 7 Exercise 6.1 Question 23 (Page No. 396)
How many positive integers between $100$ and $999$ inclusive are divisible by $7?$ are odd? have the same three decimal digits? are not divisible by $4?$ are divisible by $3$ or $4?$ are not divisible by either $3$ or $4?$ are divisible by $3$ but not by $4?$ are divisible by $3$ and $4?$

How many positive integers between $100$ and $999$ inclusiveare divisible by $7?$are odd?have the same three decimal digits?are not divisible by $4?$are divisible by $3$ ...

Mayan Shankar

11.7k views

Mayan Shankar commented Feb 14

0 votes

2 answers

27

Kenneth Rosen Edition 7 Exercise 6.3 Question 26 (Page No. 414)
Thirteen people on a softball team show up for a game. How many ways are there to choose $10$ players to take the field? How many ways are there to assign the $10$ positions by selecting players from the $13$ people who show ... ways are there to choose $10$ players to take the field if at least one of these players must be a woman?

Thirteen people on a softball team show up for a game.How many ways are there to choose $10$ players to take the field?How many ways are there to assign the $10$ position...

Priyam Garg

4.6k views

Priyam Garg answer edited Feb 11

0 votes

0 answers

28

Combinatorics & Probability
A rumor is spread randomly among a group of 10 people by successively having one person call someone, who calls someone, and so on. A person can pass the rumor on to anyone except the individual who just called. (a) By how many different paths can a rumor ... in $N$ calls? (c) What is the probability that if $A$ starts the rumor, then $A$ receives the third calls?

A rumor is spread randomly among a group of 10 people by successively having one person call someone, who calls someone, and so on. A person can pass the rumor on to anyo...

Debargha Mitra Roy

166 views

Debargha Mitra Roy asked Feb 8

23 votes

4 answers

29

GATE IT 2007 | Question: 76
Consider the sequence $\langle x_n \rangle , \: n \geq 0$ defined by the recurrence relation $x_{n+1} = c . x^2_n -2$, where $c > 0$. Suppose there exists a non-empty, open interval $(a, b)$ such that for all $x_0$ satisfying $a < x_0 < b$, the ... sequence converges to the value? $\frac{1+\sqrt{1+8c}}{2c}$ $\frac{1-\sqrt{1+8c}}{2c}$ $2$ $\frac{2}{2c-1}$

Consider the sequence $\langle x_n \rangle , \: n \geq 0$ defined by the recurrence relation $x_{n+1} = c . x^2_n -2$, where $c 0$.Suppose there exists a non-empty, open...

arun1504

5.5k views

arun1504 commented Feb 7

6 votes

2 answers

30

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 56
The coefficient of $x^6$ in the expansion of $A(x)$ is, where $ A(x)=\frac{x(1+x)}{(1-x)^3} $

The coefficient of $x^6$ in the expansion of $A(x)$ is, where$$A(x)=\frac{x(1+x)}{(1-x)^3}$$

squirrel69

560 views

squirrel69 answered Feb 6

tags	tag:apple
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title	title:apple
content	content:apple
exclude	-tag:apple
force match	+apple
views	views:100
score	score:10
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is accepted	isaccepted:true
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