COUNTING: Solutions Manual 2nd Edition by Koh Khee Meng Tay Eng Guan

Author:Koh Khee Meng, Tay Eng Guan
Language: eng
Format: epub
ISBN: 9789814401944
Publisher: Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.

16.5 Let Fn denote the number of pairs of rabbits at the end of the nth month, where n ≥ 1, as given in Fibonacci’s problem of rabbits. Show that

Solution Let us state clearly the chronology of the rabbits. A rabbit is born at the beginning of a month. At the beginning of the next month, it is deemed to be one month old. It (together with its partner) will give birth at the beginning of the next month (after a gestation period of a month) when it is deemed to be two months old.

Observe first that is also the number of rabbits that are at least i months old at the beginning of the nth month. The rabbits which make up the number Fn can be divided into two groups - those which existed the previous month (at least one month old at the beginning of the n-th month) and those which are babies (born at the beginning of the month). The number in the first group is Fn − 1. The babies from the second group all had parents that were at least two months old at the beginning of the month. Since a pair of babies came from a pair of parents, the number of babies is the same as the number of parents, i.e. for n ≥ 3.

16.6 Find a recurrence relation and initial conditions for the number of binary sequences of length n with no consecutive 0’s.

Solution Let sn be the number of n-digit binary sequences without the subsequence 00.

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