discrete structures I , thank you some answer are given some are not.. also i ha
ID: 3010700 • Letter: D
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discrete structures I , thank you some answer are given some are not.. also i have written ONLY HAVE TO DO thank you so much..
5.2.5: Counting ways to line up for a family photo.
A family lines up for a photograph. In each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters?
The family consists of two parents, two daughters and two sons.
The family consists of two parents, three daughters and four sons.
5.5.2: Counting telephone numbers.
At a certain university in the U.S., all phone numbers are 7-digits long and start with either 824 or 825.
How many different phone numbers are possible?
There are two choices for the first three digits (824 or 825). Each of the remaining 4 digits can be any one of the ten digits, so there are 104 ways to pick the last four digits. The total number of 7-digit phone numbers that start with 824 or 825 is 2·104.
How many different phone numbers are there in which the last four digits are all different?
5.5.3: Lining up a wedding party for a photo.
Ten members of a wedding party are lining up in a row for a photograph.
How many ways are there to line up the ten people?
A line-up of the ten people is just a permutation of ten distinct items/people. There are 10! ways to order ten people.
How many ways are there to line up the ten people if the groom must be to the immediate left of the bride in the photo?
5.6.2: Permutations and combinations from a set of letters.
Define the set S = {a, b, c, d, e, f, g}.
Give an example of a 4-permutation from the set S.
Give an example of a 4-subset from the set S.
{a, b, f, g}
How many subsets of S have exactly four elements?
There are (74)(74) ways to select a subset of 4 items from a set of 7 distinct items.
How many subsets of S have either three or four elements?
5.6.3: Counting bit strings.
How many 10-bit strings are there subject to each of the following restrictions?
(c)
The string starts with 001 or 10.
(f)
The string has exactly six 0's and the first bit is 1.
5.6.6: Counting possible computer failures.
Suppose a network has 40 computers of which 5 fail.
How many possibilities are there for the five that fail?
There are (405)(405) ways to select a subset of 5 computers from a set of 40 computers.
Suppose that 3 of the computers in the network have a copy of a particular file. How many sets of failures wipe out all the copies of the file? That is, how many 5-subsets contain the three computers that have the file?
5.6.7: Choosing a student committee.
14 students have volunteered for a committee. Eight of them are seniors and six of them are juniors.
How many ways are there to select a committee of 5 students?
There are (145)(145) ways to select a subset of 5 students from a set of 14 students.
How many ways are there to select a committee with 3 seniors and 2 juniors?
A family lines up for a photograph. In each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters?
(a)The family consists of two parents, two daughters and two sons.
(b)The family consists of two parents, three daughters and four sons.
5.5.2: Counting telephone numbers.
At a certain university in the U.S., all phone numbers are 7-digits long and start with either 824 or 825.
(a)How many different phone numbers are possible?
SolutionThere are two choices for the first three digits (824 or 825). Each of the remaining 4 digits can be any one of the ten digits, so there are 104 ways to pick the last four digits. The total number of 7-digit phone numbers that start with 824 or 825 is 2·104.
(b) HAVE to DO B ONLYHow many different phone numbers are there in which the last four digits are all different?
5.5.3: Lining up a wedding party for a photo.
Ten members of a wedding party are lining up in a row for a photograph.
(a)How many ways are there to line up the ten people?
SolutionA line-up of the ten people is just a permutation of ten distinct items/people. There are 10! ways to order ten people.
(b) HAVE TO DO B ONLYHow many ways are there to line up the ten people if the groom must be to the immediate left of the bride in the photo?
5.6.2: Permutations and combinations from a set of letters.
Define the set S = {a, b, c, d, e, f, g}.
(a) HAVE TO DO A,D ONLYGive an example of a 4-permutation from the set S.
(b)Give an example of a 4-subset from the set S.
Solution{a, b, f, g}
(c)How many subsets of S have exactly four elements?
SolutionThere are (74)(74) ways to select a subset of 4 items from a set of 7 distinct items.
(d)How many subsets of S have either three or four elements?
5.6.3: Counting bit strings.
How many 10-bit strings are there subject to each of the following restrictions?
(c)
The string starts with 001 or 10.
(f)
The string has exactly six 0's and the first bit is 1.
5.6.6: Counting possible computer failures.
Suppose a network has 40 computers of which 5 fail.
(a) HAVE TO DO B ONLYHow many possibilities are there for the five that fail?
SolutionThere are (405)(405) ways to select a subset of 5 computers from a set of 40 computers.
(b) HAVE TO DO B ONLYSuppose that 3 of the computers in the network have a copy of a particular file. How many sets of failures wipe out all the copies of the file? That is, how many 5-subsets contain the three computers that have the file?
5.6.7: Choosing a student committee.
14 students have volunteered for a committee. Eight of them are seniors and six of them are juniors.
(a)How many ways are there to select a committee of 5 students?
SolutionThere are (145)(145) ways to select a subset of 5 students from a set of 14 students.
(b) HAVE TO DO B ONLYHow many ways are there to select a committee with 3 seniors and 2 juniors?
Explanation / Answer
Question 5.2.5
A family lines up for a photograph. In each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters?
(a)The family consists of two parents, two daughters and two sons.
Total way would be:
(Mother and first daughter together) + Mother and second daughter together) - (Mother and both daughters together)
2.5! + 2.5! - 2.4! = 432 ways
(b)The family consists of two parents, three daughters and four sons.
Similarly, for this part it would be:
(2.8!)*3 - (2.7!)*3 - 2.6! = 210240 Ways
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