Recent questions and answers in Combinatory - GATE Overflow

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1

NIELIT 2017 July Scientist B (IT) - Section B: 14
There are four bus lines between A and B; and three bus lines between B and C The number of way a person roundtrip by bus from A to C by way of B will be $12$ $7$ $144$ $264$

answered 5 days ago in Combinatory by immanujs Active (2.9k points) | 10 views

0 votes

0 answers

2

NIELIT 2016 MAR Scientist B - Section B: 1
The number of ways to cut a six sided convex polygon whose vertices are labeled into four triangles using diagonal lines that do not cross is $13$ $14$ $12$ $11$

asked Mar 31 in Combinatory by Lakshman Patel RJIT Veteran (61.4k points) | 6 views

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0 answers

3

NIELIT 2016 MAR Scientist B - Section B: 2
The number of ways in which a team of eleven players can be selected from $22$ players including $2$ of them and excluding $4$ of them is $16\large_{C_{11}}$ $16\large_{C_{5}}$ $16\large_{C_{9}}$ $20\large_{C_{9}}$

asked Mar 31 in Combinatory by Lakshman Patel RJIT Veteran (61.4k points) | 9 views

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0 answers

4

NIELIT 2016 DEC Scientist B (IT) - Section B: 34
Mala has a colouring book in which each english letter is drawn two times. She wants to point each of these $52$ prints with one of $k$ colours, such that the colour-pairs used to colour ay two letters are different. Both prints of a letter ... with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$

asked Mar 31 in Combinatory by Lakshman Patel RJIT Veteran (61.4k points) | 6 views

+1 vote

1 answer

5

ISI2016-PCB-CS-8
Consider all possible trees with $n$ nodes. Let $k$ be the number of nodes with degree greater than 1 in a given tree. What is the maximum possible value of $k$? Justify your answer. Consider $2n$ committees, each having at least $2n$ persons, formed from a group of $4n$ persons. Prove that there exists at least one person who belongs to at least $n$ committees.

answered Mar 18 in Combinatory by Falahamin (21 points) | 48 views

0 votes

1 answer

6

ISI2016-MMA-25
A integer is said to be a $\textbf{palindrome}$ if it reads the same forward or backward. For example, the integer $14541$ is a $5$-digit palindrome and $12345$ is not a palindrome. How many $8$-digit palindromes are prime? $0$ $1$ $11$ $19$

answered Mar 18 in Combinatory by Falahamin (21 points) | 25 views

+4 votes

6 answers

7

GATE2020-CS-42
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ______.

answered Mar 15 in Combinatory by felics moses 1 (139 points) | 1.5k views

+1 vote

2 answers

8

ISI2017-MMA-26
Let $n$ be the number of ways in which $5$ men and $7$ women can stand in a queue such that all the women stand consecutively. Let $m$ be the number of ways in which the same $12$ persons can stand in a queue such that exactly $6$ women stand consecutively. Then the value of $\frac{m}{n}$ is $5$ $7$ $\frac{5}{7}$ $\frac{7}{5}$

answered Mar 9 in Combinatory by kraken_wizard (41 points) | 99 views

0 votes

1 answer

9

website
There is 4 coins 1 paisa, 5 paise, 10 paise, 25 paise using these coins we have to make 50 paisa how many combination can we make ?

answered Mar 1 in Combinatory by smsubham Boss (16.8k points) | 46 views

+3 votes

2 answers

10

Rosen 7e Exercise-6.5 question 45.b page 433
How many ways can n books be placed on k distinguishable shelves if no two books are the same, and the positions of the books on the shelves matter?

answered Mar 1 in Combinatory by smsubham Boss (16.8k points) | 216 views

+17 votes

3 answers

11

TIFR2012-A-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.

answered Feb 15 in Combinatory by Pratyush Priyam Kuan Active (1.1k points) | 1.4k views

+30 votes

9 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 Feb 15 in Combinatory by Pratyush Priyam Kuan Active (1.1k points) | 3.5k views

+16 votes

6 answers

13

GATE2003-34
$m$ 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 $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$

answered Feb 15 in Combinatory by Pratyush Priyam Kuan Active (1.1k points) | 2.8k views

+22 votes

4 answers

14

GATE2002-13
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 the answer ... integer $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.

answered Feb 15 in Combinatory by Pratyush Priyam Kuan Active (1.1k points) | 1.6k views

+9 votes

12 answers

15

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

answered Feb 14 in Combinatory by Pratyush Priyam Kuan Active (1.1k points) | 6.1k views

+44 votes

10 answers

16

GATE2016-1-27
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 __________.

answered Feb 2 in Combinatory by Lakshman Patel RJIT Veteran (61.4k points) | 8.5k views

0 votes

1 answer

17

sheldon ross
Three balls are to be randomly selected without replacement from an urn containing $20$ balls numbered $1$ through $20$. If we bet that at least one of the balls that are drawn has a number as large as or larger than $17$, what is the probability that we win the bet?

answered Jan 31 in Combinatory by Ayan Kumar Pahari Junior (553 points) | 132 views

+41 votes

12 answers

18

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

answered Jan 29 in Combinatory by Srividyaketharaju (117 points) | 10.4k views

+7 votes

10 answers

19

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

answered Jan 29 in Combinatory by shivam001 Active (1.2k points) | 702 views

+37 votes

8 answers

20

GATE2017-2-47
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .

answered Jan 29 in Combinatory by ankitgupta.1729 Boss (18.1k points) | 6.8k views

+21 votes

5 answers

21

GATE2000-5
A multiset is an unordered collection of elements where elements may repeat any number of times. The size of a multiset is the number of elements in it, counting repetitions. What is the number of multisets of size $4$ that can be constructed from n distinct elements so that at least one element occurs exactly twice? How many multisets can be constructed from n distinct elements?

answered Jan 26 in Combinatory by blackcloud Junior (675 points) | 1.8k views

+1 vote

1 answer

22

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 Jan 16 in Combinatory by suvradip das (189 points) | 245 views

+38 votes

2 answers

23

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 Jan 16 in Combinatory by Rishiryanemo (29 points) | 1.7k views

+27 votes

4 answers

24

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

answered Jan 14 in Combinatory by Kushagra गुप्ता Loyal (6k points) | 4k views

+42 votes

7 answers

25

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 Jan 13 in Combinatory by Kushagra गुप्ता Loyal (6k points) | 5k views

+17 votes

7 answers

26

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 (323 points) | 2k views

+30 votes

2 answers

27

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 (323 points) | 2.8k views

+23 votes

10 answers

28

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) | 7.7k views

+1 vote

3 answers

29

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 (927 points) | 2.9k views

+1 vote

1 answer

30

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) | 48 views

+1 vote

1 answer

31

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 (369 points) | 86 views

+10 votes

5 answers

32

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 (19.7k points) | 532 views

+31 votes

13 answers

33

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 (99 points) | 10.8k views

+1 vote

2 answers

34

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 (825 points) | 61 views

+2 votes

1 answer

35

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 Loyal (6k points) | 246 views

+1 vote

2 answers

36

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 Loyal (5.3k points) | 55 views

+3 votes

0 answers

37

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 (25.5k points) | 248 views

+2 votes

2 answers

38

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 $\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \left( 1+\dfrac{C_{n-1}}{C_n} \right)$ 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!} $

asked Sep 23, 2019 in Combinatory by Arjun Veteran (436k points) | 206 views

+2 votes

3 answers

39

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

asked Sep 23, 2019 in Combinatory by Arjun Veteran (436k points) | 102 views

+2 votes

0 answers

40

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, 2019 in Combinatory by Arjun Veteran (436k points) | 115 views

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