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241
Kenneth Rosen Edition 7 Exercise 6.5 Question 6 (Page No. 432)
How many ways are there to select five unordered elements from a set with three elements when repetition is allowed?
How many ways are there to select five unordered elements from a set with three elements when repetition is allowed?
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242
Kenneth Rosen Edition 7 Exercise 6.5 Question 5 (Page No. 432)
How many ways are there to assign three jobs to five employees if each employee can be given more than one job?
How many ways are there to assign three jobs to five employees if each employee can be given more than one job?
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245
Kenneth Rosen Edition 7 Exercise 6.5 Question 2 (Page No. 432)
In how many different ways can five elements be selected in order from a set with five elements when repetition is allowed?
In how many different ways can five elements be selected in order from a set with five elements when repetition is allowed?
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246
Kenneth Rosen Edition 7 Exercise 6.5 Question 1 (Page No. 432)
In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?
In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?
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255
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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257
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.]
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 th...
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258
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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259
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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260
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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261
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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262
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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263
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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269
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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