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Recent questions tagged binomial-theorem

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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.
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.
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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.
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.
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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.
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 a...
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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 $\displaystyle{}\sum_{k=1}^{n} \binom{n}{k}\binom{n}{k − 1} = \dfrac{\binom{2n + 2}{n + 1}}{2} − \binom{2n}{n}.$
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{...
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Kenneth Rosen Edition 7 Exercise 6.4 Question 25 (Page No. 422)
Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
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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 $\binom{p}{k} .$
Show that if $p$ is a prime and $k$ is an integer such that $1 \leq k \leq p − 1,$ then $p$ divides $\binom{p}{k} .$
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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.
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...
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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 $\binom{n}{k} \leq \frac{n^{k}}{2^{k−1}}.$
Show that if $n$ and $k$ are integers with $1 \leq k \leq n,$ then $\binom{n}{k} \leq \frac{n^{k}}{2^{k−1}}.$
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