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Recent questions tagged binomial-theorem

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31

Kenneth Rosen Edition 7 Exercise 6.4 Question 15 (Page No. 421)
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$

Lakshman Patel RJIT asked in Combinatory Apr 30, 2020

by Lakshman Patel RJIT

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32

Kenneth Rosen Edition 7 Exercise 6.4 Question 14 (Page No. 421)
Show that if $n$ is a positive integer, then $1 = \binom{n}{0}<\binom{n}{1}<\dots < \binom{n}{\left \lfloor n/2 \right \rfloor} = \binom{n}{\left \lceil n/2 \right \rceil}>\dots \binom{n}{n-1}>\binom{n}{n}=1.$

Lakshman Patel RJIT asked in Combinatory Apr 30, 2020

by Lakshman Patel RJIT

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33

Kenneth Rosen Edition 7 Exercise 6.4 Question 13 (Page No. 421)
What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$

Lakshman Patel RJIT asked in Combinatory Apr 30, 2020

by Lakshman Patel RJIT

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34

Kenneth Rosen Edition 7 Exercise 6.4 Question 12 (Page No. 421)
The row of Pascal’s triangle containing the binomial coefficients $\binom{10}{k},\: 0 \leq k \leq 10, \:\text{is:}\: 1\:\: 10\:\: 45\:\: 120\:\: 210\:\: 252\:\: 210\:\: 120\:\: 45\:\: 10\:\: 1$ Use Pascal’s identity to produce the row immediately following this row in Pascal’s triangle.

Lakshman Patel RJIT asked in Combinatory Apr 30, 2020

by Lakshman Patel RJIT

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35

Kenneth Rosen Edition 7 Exercise 6.4 Question 11 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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36

Kenneth Rosen Edition 7 Exercise 6.4 Question 10 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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37

Kenneth Rosen Edition 7 Exercise 6.4 Question 9 (Page No. 421)
What is the coefficient of $x^{101}y^{99}$ in the expansion of $(2x − 3y)^{200}?$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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38

Kenneth Rosen Edition 7 Exercise 6.4 Question 8 (Page No. 421)
What is the coefficient of $x^{8}y^{9}$ in the expansion of $(3x + 2y)^{17}?$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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39

Kenneth Rosen Edition 7 Exercise 6.4 Question 7 (Page No. 421)
What is the coefficient of $x^{9}\:\text{in}\: (2 − x)^{19}?$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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40

Kenneth Rosen Edition 7 Exercise 6.4 Question 6 (Page No. 421)
What is the coefficient of $x^{7}\:\text{in}\: (1 + x)^{11}?$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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41

Kenneth Rosen Edition 7 Exercise 6.4 Question 5 (Page No. 421)
How many terms are there in the expansion of $(x + y)^{100}$ after like terms are collected?

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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42

Kenneth Rosen Edition 7 Exercise 6.4 Question 4 (Page No. 421)
Find the coefficient of $x^{5}y^{8}\:\text{in}\: (x + y)^{13}.$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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43

Kenneth Rosen Edition 7 Exercise 6.4 Question 3 (Page No. 421)
Find the expansion of $(x + y)^{6}.$

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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44

Kenneth Rosen Edition 7 Exercise 6.4 Question 2 (Page No. 421)
Find the expansion of $(x + y)^{5}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

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45

Kenneth Rosen Edition 7 Exercise 6.4 Question 1 (Page No. 421)
Find the expansion of $(x + y)^{4}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.

Lakshman Patel RJIT asked in Combinatory Apr 29, 2020

by Lakshman Patel RJIT

663 views

2 votes

2 answers

46

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!} $

Arjun asked in Combinatory Sep 23, 2019

by Arjun

603 views

2 votes

3 answers

47

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

Arjun asked in Combinatory Sep 23, 2019

by Arjun

479 views

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48

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

Arjun asked in Combinatory Sep 23, 2019

by Arjun

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49

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

Arjun asked in Combinatory Sep 23, 2019

by Arjun

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50

ISI2015-DCG-18
The value of $(1.1)^{10}$ correct to $4$ decimal places is $2.4512$ $1.9547$ $2.5937$ $1.4512$

gatecse asked in Quantitative Aptitude Sep 18, 2019

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51

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

gatecse asked in Combinatory Sep 18, 2019

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52

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

gatecse asked in Combinatory Sep 18, 2019

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53

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

gatecse asked in Combinatory Sep 18, 2019

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2 votes

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54

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

gatecse asked in Combinatory Sep 18, 2019

408 views

2 votes

1 answer

55

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

gatecse asked in Combinatory Sep 18, 2019

331 views

1 vote

1 answer

56

ISI2016-MMA-14
The number of terms independent of $x$ in the binomial expansion of $\left(3x^2 + \dfrac{1}{x}\right)^{10}$ is $0$ $1$ $2$ $5$

go_editor asked in Combinatory Sep 13, 2018

326 views

5 votes

2 answers

57

BINOMIAL DISTRIBUTION
Consider an unbiased cubic dice with opposite faces coloured identically and each face coloured red, blue or green such that each colour appears only two times on the dice. If the dice is thrown thrice, the probability of obtaining red colour on top face of the dice at least twice is---- I am getting 0.75 can anyone confirm this ?

junaid ahmad asked in Probability Oct 6, 2017

by junaid ahmad

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58

Test-Book Live 2017
Which power of x has the greatest coefficient in the expansion of (1+1/2 x)^10 ?

Sarvottam Patel asked in Mathematical Logic Jan 18, 2017

by Sarvottam Patel

399 views

4 votes

3 answers

59

TIFR CSE 2016 | Part A | Question: 13
Let $n \geq 2$ be any integer. Which of the following statements is not necessarily true? $\begin{pmatrix} n \\ i \end{pmatrix} = \begin{pmatrix} n-1 \\ i \end{pmatrix} + \begin{pmatrix} n-1 \\ i-1 \end{pmatrix}, \text{ where } 1 \leq i \leq n-1$ $n!$ divides the ... $ i \in \{1, 2, \dots , n-1\}$ If $n$ is an odd prime, then $n$ divides $2^{n-1} -1$

go_editor asked in Combinatory Dec 28, 2016

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5 votes

4 answers

60

Time complexity and output
#include <stdio.h> #define N 3 int main() { int array[N] = {1,2,3}; int i,j; for ( i=1; i<(1<<N); i++) { for( j=0; j<N; j++) { if((1<<j)&i) { printf("%d", array[j]); } } printf("\n"); } return 0 ... $N = n \;\; , n \; \text{ is a positive integer }$ ? B. What is the output? C. What will be the complexity when $N$ is large.

dd asked in Programming Dec 17, 2016

by dd

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