Problem Set 7: Solutions

Problem Set 7
Section 4.1
#12 How many bit strings are there of length 6 or less?
#16 How many strings are there of 4 lowercase letters that have the letter x in them?
#22 How many strings of 4 decimal digits:
a) Do not contain the same digit twice?
b) End with an even digit?
c) Have exactly 3 digits that are 9s?
#40 How many bit strings of length 7 either begin with 2 0s or end with 3 1s?
Section 4.2
a) Show that if 7 integers are selected from the first 10 positive integers, there must be at least 2
pairs of these integers with the sum 11
b) Is the conclusion in part a true if 6 integers are selected rather that 7?
#24 Show that in a group of 5 people (where any 2 people are either friends or enemies), there
are not necessarily 3 mutual friends or 3 mutual enemies.
#28 A computer network consists of 6 computers. Each computer is directly connected to at
least one of the other computers. Show that there are at least 2 computers in the network that are
directly connected to the same number of other computers.
Discrete Computational Structures
discreteps7s06 - work
Spring 2006
Page 1
Section 4.3
#10 There are 6 different candidates for governor of a state. In how many different orders can
the names of the candidates be printed on a ballot?
#14 In how many ways can a set of two positive integers less than 100 be chosen?
#28 A professor writes 40 discrete mathematics true/false questions. Of the statements in the
questions, 17 are true. If the questions can be positioned in any order, how many different
answer keys are possible?
Section 4.4
#6 What is the coefficient of x7 in (1+x)11?
#8 What is the coefficient of x8y9 in the expansion of (3x+2y)17?
#12 The row of Pascal's triangle containing the binomial coefficients c(10,k), 0<=k<=10 is:
1 10 45 120 210 252 210 120 45 10 1
Use Pascal's identity to produce the row immediately following this row in Pascal's triangle
Section 5.1
#18 What is the probability that a 5-card poker hand contains a straight flush, that is, 5 cards of
the same suit of consecutive kinds?
#26 Find the probability of selecting none of the correct 6 integers in the lotto, where the order
in which these integers are selected does not matter, from the positive integers not exceeding:
a) 40:
b) 48:
c) 56:
d) 64:
Discrete Computational Structures
discreteps7s06 - work
Spring 2006
Page 2
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