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Recent questions tagged binomial-theorem
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TIFR CSE 2022 | Part B | Question: 9
Let $n \geq 2$ be any integer. Which of the following statements is $\text{FALSE}?$ $n!$ divides the product of any $n$ consecutive integers $\displaystyle{}\sum_{i=0}^n\left(\begin{array}{c}n \\ i\end{array}\right)=2^n$ ... an odd prime, then $n$ divides $2^{n-1}-1$ $n$ divides $\left(\begin{array}{c}2 n \\ n\end{array}\right)$
admin
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Combinatory
Sep 1
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admin
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tifr2022
combinatory
binomial-theorem
1
vote
1
answer
2
TIFR CSE 2021 | Part A | Question: 10
Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$
soujanyareddy13
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Probability
Mar 25, 2021
by
soujanyareddy13
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tifr2021
probability
binomial-theorem
1
vote
1
answer
3
TIFR CSE 2021 | Part A | Question: 12
How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$'s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $1365=2^{10} + 2^{8}+2^{6}+2^{4}+2^{2}+2^{0}.)$ ... $\binom{11}{4}+\binom{9}{3}+\binom{7}{2}+\binom{5}{1}$ $1024$
soujanyareddy13
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Digital Logic
Mar 25, 2021
by
soujanyareddy13
289
views
tifr2021
digital-logic
number-representation
binomial-theorem
0
votes
0
answers
4
Kenneth Rosen Edition 7 Exercise 6.4 Question 39 (Page No. 422 - 423)
Determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed. [Hint: Looking at Pascal's triangle will be helpful. Although infinitely many sequences start with a specified set of terms, ... $1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825,\dots$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
353
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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5
Kenneth Rosen Edition 7 Exercise 6.4 Question 38 (Page No. 422)
Give a combinatorial proof that if n is a positive integer then $\displaystyle\sum_{k = 0}^{n} k^{2} \binom{n} {k} = n(n + 1)2^{n−2}.$ [Hint: Show that both sides count the ways to select a subset of a set of ... necessarily distinct elements from this subset. Furthermore, express the right-hand side as $n(n − 1)2^{n−2} + n2^{n−1}.]$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
88
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
6
Kenneth Rosen Edition 7 Exercise 6.4 Question 37 (Page No. 422)
Use question $33$ to prove the hockeystick identity from question $27.$ [Hint: First, note that the number of paths from $(0, 0)\: \text{to}\: (n + 1,r)$ equals $\binom{n + 1 + r}{r}.$ Second, count the number of paths by summing the number of these paths that start by going $k$ units upward for $k = 0, 1, 2,\dots,r.]$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
111
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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7
Kenneth Rosen Edition 7 Exercise 6.4 Question 36 (Page No. 422)
Use question $33$ to prove Pascal’s identity. [Hint: Show that a path of the type described in question $33$ from $(0, 0)\: \text{to}\: (n + 1 − k, k)$ passes through either $(n + 1 − k, k − 1)\: \text{or} \:(n − k, k),$ but not through both.]
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
93
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
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0
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Kenneth Rosen Edition 7 Exercise 6.4 Question 35 (Page No. 422)
Use question $33$ to prove Theorem $4.$ [Hint: Count the number of paths with n steps of the type described in question $33.$ Every such path must end at one of the points $(n − k, k)\:\text{for}\: k = 0, 1, 2,\dots,n.]$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
87
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
9
Kenneth Rosen Edition 7 Exercise 6.4 Question 34 (Page No. 422)
Use question $33$ to give an alternative proof of Corollary $2$ in Section $6.3,$ which states that $\binom{n}{k} = \binom{n}{n−k} $ whenever $k$ is an integer with $0 \leq k \leq n.[$Hint: Consider the number of paths of the type described in question $33$ ... $(0, 0)\: \text{to}\:(k, n − k).]$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
123
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
10
Kenneth Rosen Edition 7 Exercise 6.4 Question 33 (Page No. 422)
In this exercise we will count the number of paths in the $xy$ plane between the origin $(0, 0)$ and point $(m, n),$ where $m$ and $n$ are nonnegative integers, such that each path is made up of a series of steps, where each ... move one unit upward. Conclude from part $(A)$ that there are $\binom{m + n}{n}$ paths of the desired type.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
185
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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11
Kenneth Rosen Edition 7 Exercise 6.4 Question 32 (Page No. 422)
Prove the binomial theorem using mathematical induction.
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
105
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
proof
0
votes
0
answers
12
Kenneth Rosen Edition 7 Exercise 6.4 Question 31 (Page No. 422)
Show that a nonempty set has the same number of subsets with an odd number of elements as it does subsets with an even number of elements.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
103
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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13
Kenneth Rosen Edition 7 Exercise 6.4 Question 30 (Page No. 422)
Give a combinatorial proof that $\displaystyle{}\sum_{k = 1}^{n} k \binom{n}{k}^{2} = n \binom{2n−1}{n−1}.$ [Hint: Count in two ways the number of ways to select a committee, with $n$ members ... of $n$ mathematics professors and $n$ computer science professors, such that the chairperson of the committee is a mathematics professor.]
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
114
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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14
Kenneth Rosen Edition 7 Exercise 6.4 Question 29 (Page No. 422)
Give a combinatorial proof that $\displaystyle{}\sum_{k = 1}^{n} k \binom{n}{k} = n2^{n−1}.$ [Hint: Count in two ways the number of ways to select a committee and to then select a leader of the committee.]
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
71
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
15
Kenneth Rosen Edition 7 Exercise 6.4 Question 28 (Page No. 422)
Show that if $n$ is a positive integer, then $\binom{2n}{2} = 2\binom{n}{2} + n^{2} $ using a combinatorial argument. by algebraic manipulation.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
87
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
1
vote
0
answers
16
Kenneth Rosen Edition 7 Exercise 6.4 Question 27 (Page No. 422)
Prove the hockeystick identity $\displaystyle{}\sum_{k=0}^{r} \binom{n + k}{k} = \binom{n + r + 1}{r}$ whenever $n$ and $r$ are positive integers, using a combinatorial argument. using Pascal’s identity.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
119
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
17
Kenneth Rosen Edition 7 Exercise 6.4 Question 26 (Page No. 422)
Let $n$ and $k$ be integers with $1 \leq k \leq n.$ Show that $\displaystyle{}\sum_{k=1}^{n} \binom{n}{k}\binom{n}{k − 1} = \dfrac{\binom{2n + 2}{n + 1}}{2} − \binom{2n}{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
86
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
2
answers
18
Kenneth Rosen Edition 7 Exercise 6.4 Question 25 (Page No. 422)
Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
141
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
19
Kenneth Rosen Edition 7 Exercise 6.4 Question 24 (Page No. 422)
Show that if $p$ is a prime and $k$ is an integer such that $1 \leq k \leq p − 1,$ then $p$ divides $\binom{p}{k} .$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
100
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
20
Kenneth Rosen Edition 7 Exercise 6.4 Question 23 (Page No. 422)
Show that if $n$ and $k$ are positive integers, then $\binom{n + 1}{k} = \dfrac{(n + 1)\binom {n}{k – 1}}{k}.$ Use this identity to construct an inductive definition of the binomial coefficients.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
114
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
21
Kenneth Rosen Edition 7 Exercise 6.4 Question 22 (Page No. 422)
Prove the identity $\binom{n}{r}\binom{r}{k} = \binom{n}{k}\binom{n−k}{r−k} ,$ whenever $n, r,$ and $k$ are nonnegative integers with $r \leq n$ and $k \leq r,$ using a combinatorial argument. using an argument based on the formula for the number of $r$-combinations of a set with $n$ elements.
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
134
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
22
Kenneth Rosen Edition 7 Exercise 6.4 Question 21 (Page No. 422)
Prove that if $n$ and $k$ are integers with $1 \leq k \leq n,$ then $k \binom{n}{k} = n \binom{n−1}{k−1},$ using a combinatorial proof. [Hint: Show that the two sides of the identity count the number of ways to select ... subset.] using an algebraic proof based on the formula for $\binom{n}{r}$ given in Theorem $2$ in Section $6.3.$
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
102
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
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23
Kenneth Rosen Edition 7 Exercise 6.4 Question 20 (Page No. 421)
Suppose that $k$ and $n$ are integers with $1 \leq k<n.$ Prove the hexagon identity $\binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k} = \binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1},$ which relates terms in Pascal’s triangle that form a hexagon.
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
250
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
1
vote
0
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24
Kenneth Rosen Edition 7 Exercise 6.4 Question 19 (Page No. 421)
Prove Pascal’s identity, using the formula for $\binom{n}{r}.$
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
112
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kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
25
Kenneth Rosen Edition 7 Exercise 6.4 Question 18 (Page No. 421)
Suppose that $b$ is an integer with $b \geq 7.$ Use the binomial theorem and the appropriate row of Pascal’s triangle to find the base-$b$ expansion of $(11)^{4}_{b}$ [that is, the fourth power of the number $(11)_{b}$ in base-$b$ notation].
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
352
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
1
vote
0
answers
26
Kenneth Rosen Edition 7 Exercise 6.4 Question 17 (Page No. 421)
Show that if $n$ and $k$ are integers with $1 \leq k \leq n,$ then $\binom{n}{k} \leq \frac{n^{k}}{2^{k−1}}.$
Lakshman Patel RJIT
asked
in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
100
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
27
Kenneth Rosen Edition 7 Exercise 6.4 Question 16 (Page No. 421)
Use question $14$ and Corollary $1$ to show that if $n$ is an integer greater than $1,$ then $\binom{n}{\left \lfloor n/2 \right \rfloor}\geq \frac{2^{n}}{2}.$ Conclude from part $(A)$ that if $n$ is a positive integer, then $\binom{2n}{n}\geq \frac{4^{n}}{2n}.$
Lakshman Patel RJIT
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
331
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kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
0
answers
28
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
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
111
views
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
0
votes
1
answer
29
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
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in
Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
131
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kenneth-rosen
discrete-mathematics
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