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Massachusetts Institute of Technology
6.042J/18.062J, Spring ’10
: Mathematics for Computer Science
April 7
Prof. Albert R. Meyer
revised April 15, 2010, 1374 minutes
Problem
Set
9
Due
: April 16
Reading:
Notes Ch.
16.1
–
16.9
Problem
1.
Show that for any set of 201 positive integers less than 300, there must be two whose quotient is a
power of three (with no remainder).
Problem
2.
Answer the following questions with a number or a simple formula involving factorials and bino
mial coefﬁcients. Brieﬂy explain your answers.
(a)
How many ways are there to order the 26 letters of the alphabet so that no two of the vowels
a
,
e
,
i
,
o
,
u
appear consecutively and the last letter in the ordering is not a vowel?
Hint:
Every vowel appears to the left of a consonant.
(b)
How many ways are there to order the 26 letters of the alphabet so that there are
at least two
consonants immediately following each vowel?
(c)
In how many different ways can
2
n
students be paired up?
(d)
Two
n
digit sequences of digits 0,1,. . . ,9 are said to be of the
same type
if the digits of one are
a permutation of the digits of the other. For
n
= 8
, for example, the sequences
03088929
and
00238899
are the same type. How many types of
n
digit integers are there?
Problem
3.
Section
16.8.3
explained why it is not possible to perform a fourcard variant of the hiddencard
magic trick with one card hidden. But the Magician and her Assistant are determined to ﬁnd
a way to make a trick like this work. They decide to change the rules slightly: instead of the
Assistant lining up the three unhidden cards for the Magician to see, he will line up all four cards
with one card face down and the other three visible. We’ll call this the
facedown fourcard trick
.
For example, suppose the audience members had selected the cards
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