Recent questions and answers in Combinatory - GATE Overflow

+8 votes

5 answers

1

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 3 days ago in Combinatory by `JEET Boss (13.1k points) | 419 views

+24 votes

13 answers

2

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

answered 4 days ago in Combinatory by Praveenk99 (49 points) | 8.4k views

+37 votes

11 answers

3

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

answered 4 days ago in Combinatory by pritishc Junior (669 points) | 9k views

+1 vote

2 answers

4

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 in Combinatory by noob_coder Junior (657 points) | 26 views

+2 votes

1 answer

5

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 in Combinatory by Kushagra गुप्ता Active (1.9k points) | 198 views

+1 vote

2 answers

6

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 in Combinatory by chirudeepnamini Active (3.1k points) | 29 views

+2 votes

3 answers

7

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 in Combinatory by chirudeepnamini Active (3.1k points) | 43 views

+1 vote

1 answer

8

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 in Combinatory by Yash4444 Junior (731 points) | 17 views

+1 vote

1 answer

9

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 in Combinatory by Verma Ashish Boss (11.8k points) | 18 views

+6 votes

3 answers

10

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 in Combinatory by Optimus Prime Junior (543 points) | 2.7k views

0 votes

2 answers

11

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 in Combinatory by SaurabhKatkar (151 points) | 207 views

+2 votes

2 answers

12

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 in Combinatory by Ram Swaroop Active (4.6k points) | 303 views

0 votes

2 answers

13

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 in Combinatory by Sandeep shahu (71 points) | 59 views

+1 vote

1 answer

14

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 in Combinatory by `JEET Boss (13.1k points) | 37 views

+1 vote

1 answer

15

#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 ${\Big[\frac{2^{n+1} + (-1)^n}{3}\Big]}^2$ $(n+1)^2$ $(n-1)^3$ $(n-1)^2$

answered Nov 5 in Combinatory by techbd123 Active (3.1k points) | 129 views

+6 votes

4 answers

16

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 in Combinatory by kunal goswami Junior (533 points) | 961 views

+1 vote

0 answers

17

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 in Combinatory by Satbir Boss (21.5k points) | 161 views

+36 votes

11 answers

18

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 in Combinatory by paraskk (335 points) | 3k views

+19 votes

5 answers

19

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 in Combinatory by Asim Siddiqui 4 Active (1.3k points) | 3.1k views

+1 vote

1 answer

20

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

answered Oct 21 in Combinatory by `JEET Boss (13.1k points) | 18 views

+39 votes

6 answers

21

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 Oct 20 in Combinatory by techbd123 Active (3.1k points) | 4.4k views

+3 votes

2 answers

22

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 in Combinatory by chirudeepnamini Active (3.1k points) | 98 views

+1 vote

2 answers

23

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 in Combinatory by chirudeepnamini Active (3.1k points) | 33 views

0 votes

1 answer

24

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

answered Oct 19 in Combinatory by chirudeepnamini Active (3.1k points) | 9 views

+2 votes

3 answers

25

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 in Combinatory by chirudeepnamini Active (3.1k points) | 36 views

0 votes

1 answer

26

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 in Combinatory by anonymousgamer (11 points) | 64 views

0 votes

1 answer

27

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

answered Oct 11 in Combinatory by shivamgo (11 points) | 35 views

0 votes

1 answer

28

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 in Combinatory by techbd123 Active (3.1k points) | 28 views

+5 votes

10 answers

29

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

answered Oct 3 in Combinatory by techbd123 Active (3.1k points) | 3.2k views

+1 vote

1 answer

30

ISI2014-DCG-34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$

answered Sep 30 in Combinatory by techbd123 Active (3.1k points) | 33 views

+1 vote

2 answers

31

ISI2014-DCG-1
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of $\bigg( 1+\dfrac{C_0}{C_1} \bigg) \bigg( 1+\dfrac{C_1}{C_2} \bigg) \cdots \bigg( 1+\dfrac{C_{n-1}}{C_n} \bigg)$ is $\bigg( \frac{n+1}{n+2} \bigg) ^n$ $ \frac{n^n}{n!} $ $\bigg( \frac{n}{n+1} \bigg) ^n$ $ \frac{(n+1)^n}{n!} $

answered Sep 27 in Combinatory by STUDYGATE2019 (91 points) | 128 views

+1 vote

1 answer

32

ISI2015-MMA-1
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $ f_1(x)=(x-2)^2$ $f_{n+1}(x) = (f_n(x)-2)^2, \: \: \: n \geq 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then $a_n=4, \: b_n=-4^n$ $a_n=4, \: b_n=-4n^2$ $a_n=4^{(n-1)!}, \: b_n=-4^n$ $a_n=4^{(n-1)!}, \: b_n=-4n^2$

answered Sep 26 in Combinatory by `JEET Boss (13.1k points) | 29 views

0 votes

1 answer

33

ISI2015-MMA-6
A club with $x$ members is organized into four committees such that each member is in exactly two committees, any two committees have exactly one member in common. Then $x$ has exactly two values both between $4$ and $8$ exactly one value and this lies between $4$ and $8$ exactly two values both between $8$ and $16$ exactly one value and this lies between $8$ and $16$

answered Sep 25 in Combinatory by `JEET Boss (13.1k points) | 20 views

+1 vote

1 answer

34

ISI2014-DCG-63
If $^nC_{r-1}=36$, $^nC_r=84$ an $^nC_{r+1}=126$ then $r$ is equal to $1$ $2$ $3$ none of these

answered Sep 25 in Combinatory by `JEET Boss (13.1k points) | 15 views

0 votes

1 answer

35

ISI2014-DCG-66
Consider all possible words obtained by arranging all the letters of the word $\textbf{AGAIN}$. These words are now arranged in the alphabetical order, as in a dictionary. The fiftieth word in this arrangement is $\text{IAANG}$ $\text{NAAGI}$ $\text{NAAIG}$ $\text{IAAGN}$

answered Sep 25 in Combinatory by `JEET Boss (13.1k points) | 15 views

+1 vote

1 answer

36

ISI2014-DCG-41
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is $11$ $12$ $13$ $14$

answered Sep 24 in Combinatory by `JEET Boss (13.1k points) | 30 views

+2 votes

0 answers

37

ISI2014-DCG-32
Consider $30$ multiple-choice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$

asked Sep 23 in Combinatory by Arjun Veteran (424k points) | 50 views

0 votes

0 answers

38

ISI2015-MMA-60
Let $\sigma$ be the permutation: $\begin{array} {}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 6 & 2 & 4 & 9 & 8 & 7 & 1, \end{array}$ $I$ be the identity permutation and $m$ be the order of $\sigma$ i.e. $m=\text{min}\{\text{positive integers }n: \sigma ^n=I \}$. Then $m$ is $8$ $12$ $360$ $2520$

asked Sep 23 in Combinatory by Arjun Veteran (424k points) | 6 views

+2 votes

1 answer

39

ISI2016-DCG-24
If the letters of the word $\text{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is $8!$ $6!$ $3!6!$ None of these

answered Sep 22 in Combinatory by `JEET Boss (13.1k points) | 10 views

+1 vote

1 answer

40

ISI2018-DCG-17
The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is $4096$ $4083$ $2^{13}-1$ $2^{12}-1$

answered Sep 21 in Combinatory by kp1 (115 points) | 13 views

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