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Unformatted text preview: MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Assignment 3 Question 1 . (Binary strings and Fibonacci numbers) Suppose that the digit “0” takes one unit of time to transmit through a communication channel while the digit “1” takes two units of time. (Then the time needed to transmit a binary string with i “0”s and j “1”s equals i + 2 j units.) Let f k,n be the number of binary strings of length n that can be transmitted in k units of time. (a) Find a closed form for F ( x, y ) = summationdisplay k ≥ summationdisplay n ≥ f k,n x k y n . Find G ( x, y ) = ∂ ∂x F ( x, y ) and the number 2 − n [ y n ] G (1 , y ). What is the meaning of this number? (b) Find the number of strings g k that can be transmitted using exactly k units of time (for each string). (c) Find the number of strings that can be transmitted using at most k units of time each. Question 2 . (Binary strings with no 3 adjacent 0’s) Find a formula for the number f n of strings, with n bits, containing no 3 consecutive 0‘s. Two adjacent 0’s are allowed. Note. Using a combinatorial argument,bits, containing no 3 consecutive 0‘s....
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