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Kenneth Rosen Edition 7 Exercise 6.4 Question 19 (Page No. 421)
Lakshman Patel RJIT
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Prove Pascal’s identity, using the formula for $\binom{n}{r}.$
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Lakshman Patel RJIT
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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 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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Apr 30, 2020
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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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Apr 30, 2020
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Lakshman Patel RJIT
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kenneth-rosen
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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 $\binom{2n}{n}\geq \frac{4^{n}}{2n}.$
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