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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 +21) + F 3 n +3 by the induction hypothesis = (1 / 2)( F 3 n +2 + 2 F 3 n +31) = (1 / 2)( F 3 n +3 + F 3 n +41) by properties of Fibonacci numbers = (1 / 2)( F 3 n +51) , 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 5card 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 : 36 526 510 5 . 2...
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 Fall '07
 Graham

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