#ATTENDANCE QUIZ FOR LECTURE 13 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p13
#with an attachment called
#p12FirstLast.txt
#(e.g. p12DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 16, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 6

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH BY THE ANSWER


QUESTION #1:

Describe the problem that Euler solved regarding 7 bridges

ANSWER:

The Seven Bridges of Königsberg stated to devise a walk through the city that would cross each of those bridges once and only once. (Wikipedia)

QUESTION #2:

Draw the Graph G=[{2,3},{1,3,4},{1,2,4},{2,3}]

ANSWER:

|-----------|
1 --- 2 ----3----4
      |-----|    |
      |-----|----|


QUESTION #3:

If you toss a fair coin 2000 times what is the probability that you get exactly 1000 heads?

ANSWER:

There are 2^2000 possible outcomes. There are binomial(2000,1000) ways of getting 1000 heads, which gives you a probability of 2000!/(2^2000 * 1000!^2)

QUESTION #4:

If you roll a fair die 6000 times what is the probability that each of the possible outcomes occurs exactly 1000 times?

ANSWER:

The total number of outcomes in 6^1000.
There are 6000!/(1000!)^6 ways of getting 1000 occurrences of each outcome.
Hence, the probability is 6000!/(6^1000 * (1000!)^6).


QUESTION #5:

Pick 5 random Facebook friends.
For each of them pick 3 friends.
For each of the friends of friends pick 3 friends.
Label the people picked 1,2,...,n.
Write the graph in data structure format.

ANSWER:

V={Alex, Dan, Arjun, Gaurav, Matt, John-Paul, Carl, Greg, Katherine, Jessica, Anna, Anand, Anwar, Doug, Max, Jasmine, Shaan, Can, Joseph, Craig, Kenny, Todd, David, Andrew, Eric, Scott}

G=[{Gaurav, Matt, John-Paul}, {Carl, Greg, Katherine}, {Jessica, Anna, Anand}, {Anwar, Doug, Max}, {Jasmine, Shaan, Can}, {Alex, John-Paul, Shaan}, {Carl, Jessica, Anna}, {Anand, Anwar, Joseph}, {Doug, Max, Jasmine}, {Can, John-Paul, Jessica}, {Anna, Max, Joseph}, {Todd, Andrew, David}, {Scott, Craig, Kenny}, {Doug, Max, Anand}, {Katherine, Eric, Scott}, {Max, Doug, Craig}}


QUESTION #6:

Look at the cities that border Piscataway and for each of them those border them and again until you get too Princeton.

(i) Construct the Graph
(ii) Find the number of paths from Piscataway to Princeton
(iii) Find the set using Paths/Using NuPaths

ANSWER:

(i)
V={Piscataway, Edison, New Brunswick, Montgomery, Woodship Township, East Brunswick, South Brunswick, South Plainfield, Princeton}
G=[{Edison, New Brunswick, Montgomery, South Plainfield}, {New Brunswick, Woodship Township}, {East Brunswick, South Brunswick, Edison, Montgomery}, {East Amwell, Princeton, South Brunswick}]

(ii)
The paths are:
Piscataway -> Edison -> New Brunswick -> Montgomery -> Princeton
Piscataway -> Edison -> New Brunswick -> South Brunswick -> Princeton
Piscataway -> Edison -> New Brunswick -> South Brunswick -> WWP -> Princeton
Piscataway -> Edison -> New Brunswick -> East Brunswick -> South Brunswick -> Princeton
Piscataway -> Montgomery -> Princeton
Piscataway -> Montgomery -> East Amwell -> Princeton

(iii)
Running:
Paths(G, 1, 9, 5) we get:
{1, 2, 3, 4, 9},
{1, 2, 3, 7, 9},

Running NuPath(G,1,9,5) we get:
2