Mid1_sol

# Mid1_sol - n = 0 and 1 for example Assume it is true for...

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CSE 21 Midterm #1 October 20, 2011 1. (a) How many rearrangements using all the letters of the word SCHWARZENEGGER are there? Answer : 14! 3!2!2! = 3632428800 1.(b) How many rearrangements using only 13 of the letters of the word SCHWARZENEGGER are there? Answer : Same as (a) , since there is a 1 - 1 correspondence between 13 letter words and 14 letter words (just remove the ﬁrst letter, for example). 2.(a) In how many ways can 4 iPods and 3 iPads be distributed to 5 students if there are no restrictions? Answer : ( 8 4 )( 7 4 ) = 2450. 2.(b) Suppose in part (a) that no student can get more than one of the same item (i.e, no one gets 2 iPods or 2 iPads). Now how many ways are there? Answer : ( 5 4 )( 5 3 ) = 50. 3. Find explicit solutions to the following two recurrences: (a) x ( n + 2) = 4 x ( n + 1) - 3 x ( n ) , x (0) = 0 ,x (1) = 1; Answer : x ( n ) = (1 / 2)(3 n - 1). (b) y ( n + 2) = 3 y ( n + 1) - 4 y ( n ) , y (0) = 0 ,y (1) = 1; Answer : y ( n ) = 1 - 7 ± ( 3+ - 7 2 ² n - ± ( 3 - - 7 2 ² n 4. Let F n be our old friends, the Fibonacci numbers, deﬁned by: F 0 = 0 ,F 1 = 1 and F n +2 = F n +1 + F n for n 0. Show that n k =0 F 3 k = 1 2 ( F 3 n +2 - 1) for n 0. 1

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Proof by induction : Statement is true for
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Unformatted text preview: n = 0 and 1, for example. Assume it is true for some value n . Then n +1 X k =1 F 3 k = (1 / 2)( F 3 n +2-1) + F 3 n +3 by the induction hypothesis = (1 / 2)( F 3 n +2 + 2 F 3 n +3-1) = (1 / 2)( F 3 n +3 + F 3 n +4-1) by properties of Fibonacci numbers = (1 / 2)( F 3 n +5-1) , which completes the induction step. Could also prove this directly by using the explicit formula for F n , which would then involve summing a geometric series. 5. How many 5-card hands from an ordinary deck of 52 cards have at least 4 cards of the same suit? Answer : 4 ( 13 5 ) + 4 ( 13 4 ) · 39 = 116688. 6. A password is to be made up of 5 symbols taken from the set of 26 letters { a,b,. ..,z } and 10 numbers { , 1 ,..., 9 } . However, each password must contain at least one letter and at least one number . How many possible passwords are there? Answer : 36 5-26 5-10 5 . 2...
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Mid1_sol - n = 0 and 1 for example Assume it is true for...

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