91Ó°ÊÓ

Find the number of positive integers not exceeding 100 that are not divisible by 5 or by 7 .

The Lucas numbers satisfy the recurrence relation $$ L_{n}=L_{n-1}+L_{n-2} $$ and the initial conditions \(L_{0}=2\) and \(L_{1}=1\) a) Show that \(L_{n}=f_{n-1}+f_{n+1}\) for \(n=2,3, \ldots,\) where \(f_{n}\) is the \(n\) th Fibonacci number. b) Find an explicit formula for the Lucas numbers.

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time. b) What are the initial conditions? c) In how many ways can this person climb a flight of eight stairs?

Find the number of positive integers not exceeding 1000 that are not divisible by \(3,17,\) or \(35 .\)

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one, two, or three stairs at a time. b) What are the initial conditions? c) In how many ways can this person climb a flight of eight stairs? A string that contains only \(0 \mathrm{s}, 1 \mathrm{s},\) and 2 \(\mathrm{s}\) is called a ternary string.

How many derangements are there of a set with seven elements?

Use generating functions to determine the number of different ways 10 identical balloons can be given to four children if each child receives at least two balloons.

a) Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s. b) What are the initial conditions? c) How many ternary strings of length six do not contain two consecutive 0s?

What is the probability that none of 10 people receives the correct hat if a hatcheck person hands their hats back randomly?

Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures.