# HW3 - (c Suppose we only require that each of the ﬁve...

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CSE21 FA11 Homework #3 (10/10/11) 3.1. Prove by induction that n k =1 k 2 = n ( n +1)(2 n +1) 6 . 3.2. It is desired to form three committees from a group of 3 men and 3 women. (a) In how many ways can this be done? (b) Suppose that each of the committees must contain at least one man and at least one women. Now how many different committees are possible? 3.3. A collection of 5 red jellybeans and 10 blue jellybeans is to be dis- tributed to a group of 3 CSE majors and 2 ECE majors. (a) In how many ways can this be done if there are no restrictions? (b) Suppose that every CSE major must get at least one red jellybean, and each ECE major must get at least two blue jelly beans. Now how many ways are there of distributing the jellybeans?

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Unformatted text preview: (c) Suppose we only require that each of the ﬁve people gets at least one jellybean (of some color) but that no CSE major gets more than two blue jellybeans? Now how many ways are possible? 3.4 Solve the following linear recurrences: (a) x n +1 = 3 x n-2 , x 1 = 2; (b) x n +1 = 3 x n-2 , x 1 = 1; 3.5 Deﬁne the usual Fibonacci numbers by: F = 0 , F 1 = 1, and F n +2 = F n +1 + F n , n ≥ 0. Prove by induction that: (a) ∑ n k =1 F 2 k = F 2 n +1-1; (b) F n +1 F n-1 = F 2 n + (-1) n . 1 3.6 Solve the second order linear recurrences: (a) y n +2 = 2 y n +1 + y n , y = 0 , y 1 = 1; (b) y n +2 = 2 y n +1-y n , y = 0 , y 1 = 1 . 2...
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