Recent questions in Discrete Mathematics

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1

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 (21.6k points) | 168 views

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2 answers

2

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!} $

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 128 views

+1 vote

1 answer

3

ISI2014-DCG-15
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals $A_3$ $A_4$ $A_5$ $A_6$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 31 views

+2 votes

3 answers

4

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 in Combinatory by Arjun Veteran (425k points) | 43 views

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0 answers

5

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 (425k points) | 51 views

+1 vote

1 answer

6

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 $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 34 views

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1 answer

7

ISI2014-DCG-35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ p-q \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 18 views

+1 vote

1 answer

8

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 $11$ $12$ $13$ $14$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 31 views

+1 vote

1 answer

9

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

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 16 views

0 votes

1 answer

10

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 $\text{IAANG}$ $\text{NAAGI}$ $\text{NAAIG}$ $\text{IAAGN}$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 15 views

+1 vote

2 answers

11

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 $24$ $16$ $12$ $32$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 26 views

0 votes

1 answer

12

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

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 28 views

+1 vote

1 answer

13

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 $a_n=4, \: b_n=-4^n$ $a_n=4, \: b_n=-4n^2$ $a_n=4^{(n-1)!}, \: b_n=-4^n$ $a_n=4^{(n-1)!}, \: b_n=-4n^2$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 29 views

+2 votes

3 answers

14

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

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 36 views

+1 vote

2 answers

15

ISI2015-MMA-5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n-1}$ $2^{2n}$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 40 views

0 votes

1 answer

16

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 $16$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 20 views

+1 vote

2 answers

17

ISI2015-MMA-7
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when divided by $3\}$. Then the number of elements in $\mathcal{R}$ is $40$ $36$ $34$ $33$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 24 views

+1 vote

2 answers

18

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 $n^2$ $n^3$ $2^n$ $3^n$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 33 views

0 votes

1 answer

19

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!} $ $\bigg( \frac{n}{n+1} \bigg) ^n$ $\frac{(n+1)^n}{n!}$

asked Sep 23 in Combinatory by Arjun Veteran (425k points) | 9 views

0 votes

0 answers

20

ISI2015-MMA-23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ ... $f(x,A)+f(x,B)\: - f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\: - f(x,B) \mid $

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 5 views

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0 answers

21

ISI2015-MMA-31
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 9 views

0 votes

0 answers

22

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 (425k points) | 6 views

0 votes

0 answers

23

ISI2015-MMA-92
Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... is of finite order $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 9 views

0 votes

1 answer

24

ISI2015-MMA-93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 15 views

0 votes

1 answer

25

ISI2015-MMA-94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$

asked Sep 23 in Set Theory & Algebra by Arjun Veteran (425k points) | 10 views

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1 answer

26

ISI2015-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

asked Sep 18 in Combinatory by gatecse Boss (16.8k points) | 11 views

+1 vote

1 answer

27

ISI2015-DCG-24
If the letters of the word $\textbf{COMPUTER}$ 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

asked Sep 18 in Combinatory by gatecse Boss (16.8k points) | 14 views

0 votes

1 answer

28

ISI2015-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of triangle $x,$ is one-one and into one-one and onto many-one and onto many-one and into

asked Sep 18 in Set Theory & Algebra by gatecse Boss (16.8k points) | 20 views

0 votes

1 answer

29

ISI2015-DCG-35
Let $A$, $B$ and $C$ be three non empty sets. Consider the two relations given below: $\begin{array}{lll} A-(B-C)=(A-B) \cup C & & (1) \\ A – (B \cup C) = (A -B)-C & & (2) \end{array}$ Both $(1)$ and $(2)$ are correct $(1)$ is correct but $(2)$ is not $(2)$ is correct but $(1)$ is not Both $(1)$ and $(2)$ are incorrect

asked Sep 18 in Set Theory & Algebra by gatecse Boss (16.8k points) | 8 views

+1 vote

1 answer

30

ISI2015-DCG-36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$. The number of bijective functions from $X$ to $Y$ is $n^n$ $n \log_2 n$ $n^2$ $n!$

asked Sep 18 in Set Theory & Algebra by gatecse Boss (16.8k points) | 15 views

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