CS173: Discrete Mathematical Structures
Fall 2005
Homework bonus
Due 11/27/05, 8a
1. Prove the identity
!
i
k
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i=k
n
(=
n+1
k+1
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by induction.
2. Define recursively: T is a full binary tree. You will have to look up the definition
of “full binary tree,” which will involve understanding binary trees, which will
involve understanding trees.
3. Let A and B be finite sets. How many functions f are there from domain A to
codomain B so that the image of f consists of exactly two elements?
4. Suppose we have a sequence of Bernoulli trials with probability p of success on
each trial. If the random variable X is the number of successes in n trials, find the
expected value of X3. You may leave your answer in summation form.
5.
a. How many possible selections of two dozen jelly beans are there if they
come in 5 flavors: cherry, lemon, chocolate, blueberry, and mint?
b. How many if you dislike mint and will not take more than three mint jelly
beans?
6. Determine whether each of the following functions is O(n2), Ω(n2), Θ(n2), or none
of these.
a.
!
f(n)=n
2
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.
b.
!
f(n)=nlog n
.
c.
!
f(n)=n2log n
.
d.
!
f(n)=n3if n is odd
0 if n is even
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$
.