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Kenneth Rosen Edition 7 Exercise 6.4 Question 15 (Page No. 421)
Lakshman Patel RJIT
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Apr 30, 2020
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Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
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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.
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Lakshman Patel RJIT
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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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Combinatory
Apr 30, 2020
by
Lakshman Patel RJIT
122
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kenneth-rosen
discrete-mathematics
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Lakshman Patel RJIT
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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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Apr 30, 2020
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Lakshman Patel RJIT
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kenneth-rosen
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Lakshman Patel RJIT
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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 $\binom{n}{k} \leq \frac{n^{k}}{2^{k−1}}.$
Lakshman Patel RJIT
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