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Kenneth Rosen Edition 7 Exercise 6.4 Question 23 (Page No. 422)
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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.
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Kenneth Rosen Edition 7 Exercise 6.4 Question 39 (Page No. 422 - 423)
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Kenneth Rosen Edition 7 Exercise 6.4 Question 38 (Page No. 422)
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 ... necessarily distinct elements from this subset. Furthermore, express the right-hand side as $n(n − 1)2^{n−2} + n2^{n−1}.]$
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Kenneth Rosen Edition 7 Exercise 6.4 Question 37 (Page No. 422)
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 $k = 0, 1, 2,\dots,r.]$
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Kenneth Rosen Edition 7 Exercise 6.4 Question 36 (Page No. 422)
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.]
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