Recent questions in Combinatory

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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, 2019 in Combinatory by Satbir Boss (24k points) | 214 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 $\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \left( 1+\dfrac{C_{n-1}}{C_n} \right)$ 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 (431k points) | 167 views

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3 answers

3

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 (431k points) | 67 views

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0 answers

4

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 (431k points) | 78 views

+1 vote

1 answer

5

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, 2019 in Combinatory by Arjun Veteran (431k points) | 49 views

+1 vote

1 answer

6

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, 2019 in Combinatory by Arjun Veteran (431k points) | 47 views

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1 answer

7

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, 2019 in Combinatory by Arjun Veteran (431k points) | 28 views

+1 vote

1 answer

8

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, 2019 in Combinatory by Arjun Veteran (431k points) | 29 views

+1 vote

2 answers

9

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, 2019 in Combinatory by Arjun Veteran (431k points) | 38 views

+1 vote

1 answer

10

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, 2019 in Combinatory by Arjun Veteran (431k points) | 38 views

+1 vote

1 answer

11

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, 2019 in Combinatory by Arjun Veteran (431k points) | 38 views

+2 votes

3 answers

12

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, 2019 in Combinatory by Arjun Veteran (431k points) | 46 views

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1 answer

13

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, 2019 in Combinatory by Arjun Veteran (431k points) | 26 views

+1 vote

2 answers

14

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, 2019 in Combinatory by Arjun Veteran (431k points) | 46 views

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1 answer

15

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

asked Sep 23, 2019 in Combinatory by Arjun Veteran (431k points) | 16 views

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0 answers

16

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, 2019 in Combinatory by Arjun Veteran (431k points) | 19 views

0 votes

1 answer

17

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, 2019 in Combinatory by gatecse Boss (17.5k points) | 21 views

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1 answer

18

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, 2019 in Combinatory by gatecse Boss (17.5k points) | 26 views

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1 answer

19

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

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 32 views

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1 answer

20

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

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 22 views

+1 vote

1 answer

21

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

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 22 views

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1 answer

22

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 $99$ $100$ $101$ $102$

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 51 views

+1 vote

2 answers

23

ISI2018-DCG-8
A Pizza Shop offers $6$ different toppings, and they do not take an order without any topping. I can afford to have one pizza with a maximum of $3$ toppings. In how many ways can I order my pizza? $20$ $35$ $41$ $21$

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 36 views

+1 vote

1 answer

24

ISI2018-DCG-12
Let $A=\{10,11,12,13, \dots ,99\}$. How many pairs of numbers $x$ and $y$ are possible so that $x+y\geq 100$ and $x$ and $y$ belong to $A$? $2405$ $2455$ $1200$ $1230$

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 50 views

+1 vote

1 answer

25

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 $\frac{8!}{2^4}$ $\frac{8!}{2^4(4!)}$ $\frac{8!}{4!}$ $\frac{8!}{(4!)^2}$

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 35 views

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1 answer

26

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

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 32 views

+2 votes

1 answer

27

CMI2019-A-4
There are five buckets, each of which can be open or closed. An arrangement is a specification of which buckets are open and which bucket are closed. Every person likes some of the arrangements and dislikes the rest. You host a party, and amazingly, no two people on the guest list have the ... What is the maximum number of guests possible? $5^{2}$ $\binom{5}{2}$ $2^{5}$ $2^{2^{5}}$

asked Sep 13, 2019 in Combinatory by gatecse Boss (17.5k points) | 46 views

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1 answer

28

CMI2019-A-5
Let $\pi=[x_{1},x_{2},\cdots,x_{n}]$ be a permutation of $\{1,2,\cdots,n\}.$ For $k<n,$ we say that $\pi$ has its first ascent at $k$ if $x_{1}>x_{2}\cdots>x_{k}$ and $x_{k}<x_{k+1}.$ How many permutations have their first ascent at $k?$ $\binom{n}{k}-\binom{n}{(k+1)}$ $\frac{n!}{k!}-\frac{n!}{(k+1)!}$ $\frac{n!}{(k+1)!}-\frac{n!}{(k+2)!}$ $\binom{n}{(k+1)}-\binom{n}{(k+2)}$

asked Sep 13, 2019 in Combinatory by gatecse Boss (17.5k points) | 46 views

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29

CMI2019-B-3
There is a party of $n$ people. Each attendee has at most $r$ friends in the party. The friend circle of a person includes the person and all her friends. You are required to pick some people for a party game, with the restriction that at most one person is picked from each friend circle. Show that you can pick $\dfrac{n}{r^{2}+1}$ people for the game.

asked Sep 13, 2019 in Combinatory by gatecse Boss (17.5k points) | 41 views

+1 vote

1 answer

30

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

asked Sep 13, 2019 in Combinatory by gatecse Boss (17.5k points) | 54 views

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