Recent questions and answers in Combinatory - GATE Overflow

+6 votes

8 answers

1

TIFR2018-B-1
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$

answered 2 days ago in Combinatory by Lakshman Patel RJIT Veteran (58.7k points) | 557 views

+1 vote

1 answer

2

Made Easy Test Series 2019: Combinatory - Permutations And Combinations
in how many ways 6 letters can be placed in 6 envelopes such that at least 4 letters go into their corresponding envelopes ?

answered 4 days ago in Combinatory by suvradip das (149 points) | 198 views

+37 votes

2 answers

3

GATE1991-16,a
Find the number of binary strings $w$ of length $2n$ with an equal number of $1's$ and $0's$ and the property that every prefix of $w$ has at least as many $0's$ as $1's.$

answered 4 days ago in Combinatory by Rishiryanemo (19 points) | 1.6k views

+26 votes

4 answers

4

GATE2001-2.1
How many $4$-digit even numbers have all $4$ digits distinct $2240$ $2296$ $2620$ $4536$

answered 6 days ago in Combinatory by Kushagra गुप्ता Active (4k points) | 3.8k views

+39 votes

7 answers

5

GATE2005-44
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 $b \equiv d \mod 5$ $4$ $6$ $16$ $24$

answered 6 days ago in Combinatory by Kushagra गुप्ता Active (4k points) | 4.7k views

+15 votes

7 answers

6

GATE2005-50
Let $G(x) = \frac{1}{(1-x)^2} = \sum\limits_{i=0}^\infty g(i)x^i$, where $|x| < 1$. What is $g(i)$? $i$ $i+1$ $2i$ $2^i$

answered Jan 9 in Combinatory by arjuno (275 points) | 1.9k views

+28 votes

2 answers

7

GATE2003-5
$n$ 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 \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)

answered Jan 9 in Combinatory by arjuno (275 points) | 2.6k views

+16 votes

10 answers

8

GATE2018-1
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$? $\frac{3}{(1-x)^2}$ $\frac{3x}{(1-x)^2}$ $\frac{2-x}{(1-x)^2}$ $\frac{3-x}{(1-x)^2}$

answered Jan 6 in Combinatory by Çșȇ ʛấẗẻ Active (1.9k points) | 6.9k views

+1 vote

3 answers

9

ISI2019-MMA-27
A general election is to be scheduled on $5$ days in May such that it is not scheduled on two consecutive days. In how many ways can the $5$ days be chosen to hold the election? $\begin{pmatrix} 26 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 27 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 30 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 31 \\ 5 \end{pmatrix}$

answered Jan 6 in Combinatory by Navneet Singh Tomar Junior (735 points) | 2.8k views

+1 vote

1 answer

10

ISI2017-DCG-11
The coefficient of $x^6y^3$ in the expression $(x+2y)^9$ is $84$ $672$ $8$ none of these

answered Dec 24, 2019 in Combinatory by swatiraoo45# (21 points) | 20 views

+1 vote

1 answer

11

Kenneth Rosen Edition 6th Exercise 6.4 Question 39 (Page No. 442)
What is the generating function for the sequence of Fibonacci numbers?

answered Dec 11, 2019 in Combinatory by Anup dogrial (321 points) | 79 views

+25 votes

8 answers

12

GATE2004-IT-35
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $\forall i= 1,2,\ldots 5$ and each cell contains exactly one ball? $44$ $96$ $120$ $3125$

answered Dec 8, 2019 in Combinatory by Madhab Loyal (5.7k points) | 3.2k views

+8 votes

5 answers

13

TIFR2018-A-6
What is the minimum number of students needed in a class to guarantee that there are at least $6$ students whose birthdays fall in the same month ? $6$ $23$ $61$ $72$ $91$

answered Dec 3, 2019 in Combinatory by `JEET Boss (18.9k points) | 469 views

+28 votes

13 answers

14

GATE2018-46
The number of possible min-heaps containing each value from $\{1,2,3,4,5,6,7\}$ exactly once is _______

answered Dec 2, 2019 in Combinatory by Praveenk99 (83 points) | 9.6k views

+38 votes

11 answers

15

GATE2016-1-26
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.

answered Dec 2, 2019 in Combinatory by pritishc Active (1.9k points) | 9.7k views

+1 vote

2 answers

16

ISI2014-DCG-71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$

answered Nov 28, 2019 in Combinatory by noob_coder Junior (785 points) | 38 views

+2 votes

1 answer

17

Kenneth Rosen Edition 6th Exercise 6.4 Question 13 (Page No. 440)
Use Generating function to determine,the number of different ways $10$ identical balloons can be given to four children if each child receives atleast $2$ ballons? Ans given $(x^{2}+x^{3}+.........................)^{4}$ But as there is a upper ... Which one is correct? plz confirm

answered Nov 26, 2019 in Combinatory by Kushagra गुप्ता Active (4k points) | 213 views

+1 vote

2 answers

18

ISI2016-PCB-A-3
A bit string is called legitimate if it contains no consecutive zeros $, e.g., 0101110$ is legitimate, where as $10100111$ is not. Let $a_n$ denote the number of legitimate bit strings of length $n$. Define $a_0=1$. Derive a recurrence relation for $a_n ( i.e.,$ express $a_n$ in terms of the preceding $a_i's).$

answered Nov 22, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 37 views

+2 votes

3 answers

19

ISI2014-DCG-18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n-1}$ $2^n-n2^{n-1}$ $2^n$ none of these

answered Nov 20, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 65 views

+1 vote

1 answer

20

ISI2018-DCG-14
In a room there are $8$ men, numbered $1,2, \dots ,8$. These men have to be divided into $4$ teams in such a way that every team has exactly $2$ ... of such $4$-team combinations is $\frac{8!}{2^4}$ $\frac{8!}{2^4(4!)}$ $\frac{8!}{4!}$ $\frac{8!}{(4!)^2}$

answered Nov 19, 2019 in Combinatory by Yash4444 Junior (833 points) | 35 views

+2 votes

1 answer

21

ISI2016-DCG-21
The value of the term independent of $x$ in the expansion of $(1-x)^{2}(x+\frac{1}{x})^{7}$ is $-70$ $70$ $35$ None of these

answered Nov 18, 2019 in Combinatory by Verma Ashish Boss (12.9k points) | 30 views

+9 votes

3 answers

22

GATE2019-5
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n-1}$ $\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}$ Which of the above statements is/are TRUE? Only I Only II Both I and II Neither I nor II

answered Nov 14, 2019 in Combinatory by Optimus Prime Junior (631 points) | 3.1k views

0 votes

2 answers

23

MadeEasy Full Length Test 2019: Combinatory - Permutations And Combinations
The number of ways 5 letter be put in 3 letter boxes A,B,C. If letter box A must contain at least 2 letters.

answered Nov 11, 2019 in Combinatory by SaurabhKatkar (231 points) | 233 views

+2 votes

2 answers

24

Discrete Mathematics Thegatebook
how many positive integers between 50 and 100, (a) divisible by 7 (b) divisible by 11 (c) divisible by 7 and 11?

answered Nov 10, 2019 in Combinatory by Ram Swaroop Loyal (5.3k points) | 310 views

0 votes

2 answers

25

Rosen 7e Exercise-8.1 Question no-10 Page no-511
Find a recurrence relation for the number of bit strings of length n that contain the string 01.

answered Nov 9, 2019 in Combinatory by Sandeep shahu (73 points) | 69 views

+1 vote

1 answer

26

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 $(i,j+1)?$ $\binom{2m}{n}$ $\binom{m}{n}$ $\binom{m+n}{n}$ $m^{n}$

answered Nov 9, 2019 in Combinatory by `JEET Boss (18.9k points) | 52 views

+1 vote

1 answer

27

#ACE ACADEMY BOOKLET QUESTION
The solution of $\sqrt{a_n} – 2\sqrt{a_{n-1}} + \sqrt{a_{n-2}} = 0$ where $a_0 = 1$ and $a_1 = 2$ is ${\Big[\frac{2^{n+1} + (-1)^n}{3}\Big]}^2$ $(n+1)^2$ $(n-1)^3$ $(n-1)^2$ [closed]

answered Nov 5, 2019 in Combinatory by techbd123 Active (3.5k points) | 146 views

+7 votes

4 answers

28

MadeEasy Test Series: Combinatory - Permutations And Combinations
MY SOLUTION : Fix the root then next level 2 elements ( 2! possibilities) next level 4 elements( 4! possibilities) last level 2 elements ( 2! possibilities) total possibility = 2! * 4! * 2! = 2 * 24 * 2 = 96 what ... that if node of above graph is filled with these elements it satisfies max heap property a)96 b)896 c)2688 d) none

answered Nov 4, 2019 in Combinatory by kunal goswami Junior (615 points) | 1.1k views

+2 votes

0 answers

29

The Interesting combination sum problems
Find the number of possible solutions for $x,y,z$ for each the following cases. $Case\ 1.$ Case of unlimited repetition. $x + y +z = 10$ and $x \geq 0\ , y \geq 0,\ z \geq 0 $ $Case\ 2 $ Case of unlimited repetition with variable lower bounds $x + y +z = 10$ and ... variable. $x + y +z = 10$ and $8 \geq x \geq 1\ , \ 20 \geq y \geq 2 \ , 12 \geq z \geq 3\ $

asked Nov 1, 2019 in Combinatory by Satbir Boss (23.8k points) | 211 views

+39 votes

11 answers

30

GATE2014-1-49
A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. The set of all possible $1-$pennants is ${(1)}$, the set of all possible $2-$pennants is ... $(1,2)$ is not the same as the pennant $(2,1)$. The number of $10-$pennants is________

answered Oct 28, 2019 in Combinatory by paraskk (441 points) | 3.2k views

+20 votes

5 answers

31

GATE2000-1.1
The minimum number of cards to be dealt from an arbitrarily shuffled deck of $52$ cards to guarantee that three cards are from same suit is $3$ $8$ $9$ $12$

answered Oct 22, 2019 in Combinatory by Asim Siddiqui 4 Active (1.5k points) | 3.3k views

+2 votes

1 answer

32

ISI2018-DCG-4
The number of terms with integral coefficients in the expansion of $\left(17^\frac{1}{3}+19^\frac{1}{2}x\right)^{600}$ is $99$ $100$ $101$ $102$

answered Oct 21, 2019 in Combinatory by `JEET Boss (18.9k points) | 51 views

+4 votes

2 answers

33

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 $100,000$

answered Oct 19, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 107 views

+1 vote

2 answers

34

ISI2015-MMA-8
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$

answered Oct 19, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 44 views

0 votes

1 answer

35

ISI2015-MMA-9
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \ldots +C_nx^n, \: n$ being a positive integer. The value of $\left( 1+\frac{C_0}{C_1} \right) \left( 1+\frac{C_1}{C_2} \right) \cdots \left( 1+\frac{C_{n-1}}{C_n} \right)$ is $\left( \frac{n+1}{n+2} \right) ^n$ $ \frac{n^n}{n!} $ $\left( \frac{n}{n+1} \right) ^n$ $\frac{(n+1)^n}{n!}$

answered Oct 19, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 16 views

+2 votes

3 answers

36

ISI2015-MMA-4
Suppose in a competition $11$ matches are to be played, each having one of $3$ distinct outcomes as possibilities. The number of ways one can predict the outcomes of all $11$ matches such that exactly $6$ of the predictions turn out to be correct is $\begin{pmatrix}11 \\ 6 \end{pmatrix} \times 2^5$ $\begin{pmatrix}11 \\ 6 \end{pmatrix} $ $3^6$ none of the above

answered Oct 19, 2019 in Combinatory by chirudeepnamini Active (4.3k points) | 46 views

0 votes

1 answer

37

Rosen 7e Recurrence Relation Exercise-8.1 Question no-25 page no-511
How many bit sequences of length seven contain an even number of 0s? I'm trying to solve this using recurrence relation Is my approach correct? Let T(n) be the string having even number of 0s T(1)=1 {1} T(2)=2 {00, 11} T(3)=4 {001, ... add 0 to strings of length n-1 having odd number of 0s T(n)=T(n-1) Hence, we have T(n)=2T(n-1)

answered Oct 17, 2019 in Combinatory by anonymousgamer (11 points) | 73 views

0 votes

1 answer

38

Combinatorics
There are 6n flowers of one type and 3 flowers of second type, total no. Of garlands possible?

answered Oct 11, 2019 in Combinatory by shivamgo (11 points) | 38 views

+1 vote

1 answer

39

ISI2014-DCG-72
The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j \: \: {}^kC_j$ is equal to $-1$ $0$ $1$ $2^n$

answered Oct 10, 2019 in Combinatory by techbd123 Active (3.5k points) | 37 views

+7 votes

10 answers

40

GATE2019-21
The value of $3^{51} \text{ mod } 5$ is _____

answered Oct 3, 2019 in Combinatory by techbd123 Active (3.5k points) | 4.1k views

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