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Kenneth Rosen Edition 7 Exercise 6.4 Question 37 (Page No. 422)

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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)
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Kenneth Rosen Edition 7 Exercise 6.4 Question 35 (Page No. 422)
Use question $33$ to prove Theorem $4.$ [Hint: Count the number of paths with n steps of the type described in question $33.$ Every such path must end at one of the points $(n − k, k)\:\text{for}\: k = 0, 1, 2,\dots,n.]$
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