(a) Show using a story proof that k k ! + k + 1 k ! + k + 2 k ! + + n k ! = n + 1 k + 1!

Chapter 1, Problem 18

(choose chapter or problem)

(a) Show using a story proof that k k ! + k + 1 k ! + k + 2 k ! + + n k ! = n + 1 k + 1! , where n and k are positive integers with n k. This is called the hockey stick identity. Hint: Imagine arranging a group of people by age, and then think about the oldest person in a chosen subgroup. (b) Suppose that a large pack of Haribo gummi bears can have anywhere between 30 and 50 gummi bears. There are 5 delicious flavors: pineapple (clear), raspberry (red), orange (orange), strawberry (green, mysteriously), and lemon (yellow). There are 0 nondelicious flavors. How many possibilities are there for the composition of such a pack of gummi bears? You can leave your answer in terms of a couple binomial coecients, but not a sum of lots of binomial