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Recent questions tagged binomial-theorem

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1
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, each of the following lists is the start of a sequence of ... $1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825,\dots$
asked Apr 30 in Combinatory Lakshman Patel RJIT 13 views
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2
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 $n$ elements together with two not necessarily distinct elements from this subset. Furthermore, express the right-hand side as $n(n − 1)2^{n−2} + n2^{n−1}.]$
asked Apr 30 in Combinatory Lakshman Patel RJIT 7 views
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3
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 $k = 0, 1, 2,\dots,r.]$
asked Apr 30 in Combinatory Lakshman Patel RJIT 14 views
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4
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.]
asked Apr 30 in Combinatory Lakshman Patel RJIT 11 views
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5
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6
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 $33$ from $(0, 0)\: \text{to}\: (n − k, k)$ and from $(0, 0)\: \text{to}\:(k, n − k).]$
asked Apr 30 in Combinatory Lakshman Patel RJIT 8 views
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7
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 step is a move one unit to the right or ... and a $1$ represents a move one unit upward. Conclude from part $(A)$ that there are $\binom{m + n}{n}$ paths of the desired type.
asked Apr 30 in Combinatory Lakshman Patel RJIT 9 views
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10
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 from a group of $n$ mathematics professors and $n$ computer science professors, such that the chairperson of the committee is a mathematics professor.]
asked Apr 30 in Combinatory Lakshman Patel RJIT 10 views
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11
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13
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14
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17
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18
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.
asked Apr 30 in Combinatory Lakshman Patel RJIT 7 views
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19
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 a subset with $k$ elements from a set ... then an element of this subset.] using an algebraic proof based on the formula for $\binom{n}{r}$ given in Theorem $2$ in Section $6.3.$
asked Apr 30 in Combinatory Lakshman Patel RJIT 7 views
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20
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.
asked Apr 30 in Combinatory Lakshman Patel RJIT 14 views
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22
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].
asked Apr 30 in Combinatory Lakshman Patel RJIT 19 views
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24
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 $\binom{2n}{n}\geq \frac{4^{n}}{2n}.$
asked Apr 30 in Combinatory Lakshman Patel RJIT 10 views
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26
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28
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.
asked Apr 30 in Combinatory Lakshman Patel RJIT 23 views
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