21-228 Homework 1
Due Tuesday, September 11
All answers must be justied to receive credit.
1. How many ways can you distribute three dierent pieces of candy to ve
children if no child can receive more than one piece? What about if any child
can receive an
21-228 Homework 6
Due October 31, 2001, in lecture
1. Suppose a gambler enters a casino with 1 dollar, and plays a fair game.
Each time he plays the game, he wins with probability 1/2 and loses with
probability 1/2. Each time he plays, $1 is at stake. Sup
21-228 Homework 3
Due September 25, 2001
1. Remember that we have dened a relation to be a special kind of set. For
(a) or (b) either give a proof for a yes answer, or a counterexample for a no
answer (remember the counterexample must be justied too).
(a)
21-228 Homework 5 Solutions
October 20, 2001
1. Use generating functions to solve an+3 = 3an+2 3an+1 +an where
a0 = a1 = a2 = 1
The sequence is, trivially, an = 1 for all n.
We have:
an+3 xn+3 = 3
n=0
an+2 xn+3 + 3
n=0
an+1 xn+3 +
n=0
an xn+3
n=0
Therefor
21-228 Homework 5
Due October 17, 2001
1. Use generating functions to solve an+3 = 3an+2 3an+1 + an where
a0 = a1 = a2 = 1.
2. Solve the following variant of Fibonaccis problem. each matrue pair of
rabits present at the end of any given month produces thr
21-228 Homework 6 Solutions
November 1, 2001
1. Suppose a gambler enters a casino with 1 dollar, and plays a
fair game. Each time he plays the game, he wins with probability 1/2
and loses with probability 1/2. Each time he plays, $1 is at stake.
Suppose t
21-228 Exam 1
Name:
October 3, 2001
Place your name and Section letter (whichever section you are
sitting in on) on the space provided.
You have 50 minutes. Pace yourself appropriately.
Answers may be left in terms of factorials and exponents, but
not
1. A hat check person discovers that n peoples hats have been
mixed up and returns these hats to the owners at random.
(a) In how many ways can the hats be returned so that the owners
get someone elses hat?
Let N (J) for J A be the number of ways in which
1. A hat check person discovers that n peoples hats have been mixed up
and returns these hats to the owners at random.
1. (a) In how many ways can the hats be returned so that the owners get
someone elses hat?
2. (b) What proportion of the total number of
21-228 Homework 2
Due September 18, 2001
1
Suggested Exercises (not for submission!)
These are more exercise level problems that you should practice as much as
time allows.
In particular look at 1.3.19.
1.3.1 through 1.3.10 of Bogart, except #2. 1.3.19 a
21-228 Exam 2
Name:
November 7, 2001
Place your name and Section letter (whichever section you are
sitting in on) on the space provided.
You have 50 minutes. Pace yourself appropriately.
All answers must be justied to receive credit. Please
write as le
21-228 Exam 3
Name:
December 10, 2001
Place your name and Section letter (whichever section you are
sitting in on) on the space provided.
You have 50 minutes. Pace yourself appropriately.
All answers must be justied to receive credit. Please
write as l
21-228 Homework 1
Due Tuesday, September 11
All answers must be justied to receive credit.
1. How many ways can you distribute three dierent pieces of
candy to ve children if no child can receive more than one piece?
What about if any child can receive an
21-228 Homework 2
Due September 18, 2001
n
k
1. Without appealing to the formula for
nk
m .
n
k
, prove
n
m
nm
k
=
Think in terms of elements and sets. Let A be a set of size n. We want to
choose a set B of size m and C of size k. To do so, we may either
21-228 Homework 3
Due September 25, 2001
1. Remember that we have dened a relation to be a special kind
of set. For (a) or (b) either give a proof for a yes answer, or a
counterexample for a no answer (remember the counterexample
must be justied too).
(a)