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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
128
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
158
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
234
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
1.8k
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
429
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
958
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
247
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
168
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
160
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
177
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
146
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
282
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
159
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
459
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
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
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
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
isi2014-dcg
combinatory
binomial-theorem
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
isi2014-dcg
combinatory
binomial-theorem
1
vote
1
answer
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
363
views
isi2014-dcg
combinatory
binomial-theorem
summation
1
vote
1
answer
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
431
views
isi2015-mma
combinatory
binomial-theorem
0
votes
1
answer
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
by
gatecse
306
views
isi2015-dcg
quantitative-aptitude
number-system
binomial-theorem
0
votes
1
answer
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
by
gatecse
264
views
isi2015-dcg
combinatory
binomial-theorem
2
votes
1
answer
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
by
gatecse
298
views
isi2016-dcg
combinatory
binomial-theorem
1
vote
1
answer
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
by
gatecse
418
views
isi2017-dcg
combinatory
binomial-theorem
2
votes
1
answer
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
by
gatecse
408
views
isi2018-dcg
combinatory
binomial-theorem
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
by
gatecse
331
views
isi2018-dcg
combinatory
binomial-theorem
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
by
go_editor
326
views
isi2016-mma
combinatory
binomial-theorem
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
2.0k
views
probability
binomial-theorem
0
votes
1
answer
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
binomial-theorem
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
by
go_editor
643
views
tifr2016
combinatory
binomial-theorem
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
1.3k
views
time-complexity
bitwise
programming-in-c
combinatory
summation
sub-set
binomial-theorem
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