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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
permutationandcombination
+1
vote
2
answers
2
ISI2014DCG1
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of $\bigg( 1+\dfrac{C_0}{C_1} \bigg) \bigg( 1+\dfrac{C_1}{C_2} \bigg) \cdots \bigg( 1+\dfrac{C_{n1}}{C_n} \bigg)$ 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
isi2014dcg
permutationandcombination
binomialtheorem
+1
vote
1
answer
3
ISI2014DCG15
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
isi2014dcg
settheory
algebra
+2
votes
3
answers
4
ISI2014DCG18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n1}$ $2^nn2^{n1}$ $2^n$ none of these
asked
Sep 23
in
Combinatory
by
Arjun
Veteran
(
425k
points)

43
views
isi2014dcg
permutationandcombination
binomialtheorem
+2
votes
0
answers
5
ISI2014DCG32
Consider $30$ multiplechoice 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
isi2014dcg
permutationandcombination
+1
vote
1
answer
6
ISI2014DCG34
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
isi2014dcg
permutationandcombination
binomialtheorem
summation
0
votes
1
answer
7
ISI2014DCG35
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 $q$ ... $\begin{pmatrix} m \\ pq \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
asked
Sep 23
in
Set Theory & Algebra
by
Arjun
Veteran
(
425k
points)

18
views
isi2014dcg
settheory
disjointsets
+1
vote
1
answer
8
ISI2014DCG41
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
isi2014dcg
permutationandcombination
+1
vote
1
answer
9
ISI2014DCG63
If $^nC_{r1}=36$, $^nC_r=84$ 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
isi2014dcg
permutationandcombination
0
votes
1
answer
10
ISI2014DCG66
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
isi2014dcg
permutationandcombination
arrangingletters
+1
vote
2
answers
11
ISI2014DCG71
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
isi2014dcg
permutationandcombination
arrangements
circularpermutation
0
votes
1
answer
12
ISI2014DCG72
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
isi2014dcg
permutationandcombination
summation
+1
vote
1
answer
13
ISI2015MMA1
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $ f_1(x)=(x2)^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 $a_n=4, \: b_n=4^n$ $a_n=4, \: b_n=4n^2$ $a_n=4^{(n1)!}, \: b_n=4^n$ $a_n=4^{(n1)!}, \: b_n=4n^2$
asked
Sep 23
in
Combinatory
by
Arjun
Veteran
(
425k
points)

29
views
isi2015mma
recurrencerelations
nongate
+2
votes
3
answers
14
ISI2015MMA4
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
isi2015mma
permutationandcombination
+1
vote
2
answers
15
ISI2015MMA5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n1}$ $2^{2n}$
asked
Sep 23
in
Set Theory & Algebra
by
Arjun
Veteran
(
425k
points)

40
views
isi2015mma
settheory
subsets
0
votes
1
answer
16
ISI2015MMA6
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
isi2015mma
permutationandcombination
+1
vote
2
answers
17
ISI2015MMA7
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
isi2015mma
settheory
cartesianproduct
+1
vote
2
answers
18
ISI2015MMA8
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
isi2015mma
permutationandcombination
disjointsubsets
0
votes
1
answer
19
ISI2015MMA9
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \cdots +C_nx^n, \: n$ being a positive integer. The value of $\bigg( 1+\frac{C_0}{C_1} \bigg) \bigg( 1+\frac{C_1}{C_2} \bigg) \cdots \bigg( 1+\frac{C_{n1}}{C_n} \bigg)$ 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
isi2015mma
permutationandcombination
binomialtheorem
0
votes
0
answers
20
ISI2015MMA23
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
isi2015mma
settheory
functions
nongate
0
votes
0
answers
21
ISI2015MMA31
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 nonempty none of the above
asked
Sep 23
in
Set Theory & Algebra
by
Arjun
Veteran
(
425k
points)

9
views
isi2015mma
settheory
nongate
0
votes
0
answers
22
ISI2015MMA60
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
isi2015mma
permutationandcombination
0
votes
0
answers
23
ISI2015MMA92
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
isi2015mma
grouptheory
subgroups
normal
nongate
0
votes
1
answer
24
ISI2015MMA93
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
isi2015mma
grouptheory
groups
0
votes
1
answer
25
ISI2015MMA94
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
isi2015mma
groups
quotientgroups
nongate
0
votes
1
answer
26
ISI2015DCG21
The value of the term independent of $x$ in the expansion of $(1x)^2(x+\frac{1}{x})^7$ is $70$ $70$ $35$ None of these
asked
Sep 18
in
Combinatory
by
gatecse
Boss
(
16.8k
points)

11
views
isi2015dcg
permutationandcombination
binomialtheorem
+1
vote
1
answer
27
ISI2015DCG24
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
isi2015dcg
permutationandcombination
arrangements
0
votes
1
answer
28
ISI2015DCG27
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 oneone and into oneone and onto manyone and onto manyone and into
asked
Sep 18
in
Set Theory & Algebra
by
gatecse
Boss
(
16.8k
points)

20
views
isi2015dcg
functions
triangles
0
votes
1
answer
29
ISI2015DCG35
Let $A$, $B$ and $C$ be three non empty sets. Consider the two relations given below: $\begin{array}{lll} A(BC)=(AB) \cup C & & (1) \\ A – (B \cup C) = (A B)C & & (2) \end{array}$ 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
isi2015dcg
settheory
sets
+1
vote
1
answer
30
ISI2015DCG36
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
isi2015dcg
settheory
functions
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