#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
#p13FirstLast.txt
#(e.g. p13DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 20, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE:

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

1. Describe the problem that Euler solved regarding 7 bridges.
Euler solved the problem of the seven bridges of Konigsberg, where the city of Konigsberg in Prussia was surrounded by a river on all sides that connected to other areas through seven bridges. They wished to create a walk-through that would cross each of those bridges only once, and Euler proved that it was not possible.

2. Draw it on a piece of paper with a line segment representing every edge

2. If you toss a fair coin 2000 times what is the probability that you get exactly 1000 heads (and exactly 1000 tails)
2000!/(1000!1000!)

3. If you roll a fair die 6000 times what is the probability that each of the possible outcomes {1,2,3,4,5,6} occurs exactly 1000 times
6000!/(1000!^6)

4. 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 our data structure
james(1) - isabelle(6) - nicole(21), azzy(22), ben(23)
	tim(7) - johnny(24), eric(25), nina(26)
	paula(8) - clark(27), valerie(28), patricia(29)
joseph(2) - paul(9) - sally(30), sonia(31), kyle(32)
	andrew(10) - christina(33), fred(34), zach(35)
	tina(11) - lisa(36), shayna(37), teresa(38)
ryan(3) - nicole(12) - jason(39), maria(40), derek(41)
	wesley(13) - dean(42), rosa(43), orion(44)
	rita(14) - xavier(45), newton(46), ashley(47)
nick(4) - gary(15) - andrea(48), tommy(49), gabe(50)
	jerry(16) - gabrielle(51), ashton(52), trent(53)
	kaitlyn(17) - troy(54), kristy(55), kyle(56)
anna(5) - bryan(18) - andrew(57), valerie(58), gary(59)
	lisa(19) - jeremy(60) christina(61), thomas(62)
	molly(20) - jerry(63), ben(64), jonathan
{[1, 2, 1, 65], [1, 2, 3, 65], [1, 2, 12, 65], [1, 2, 22, 65], [1, 2, 24, 65], [1, 2, 25, 65], [1, 2, 27, 65], [1, 2, 29, 65], [1, 2, 33, 65], [1, 2, 38, 65], [1, 2, 47, 65], [1, 2, 50, 65], [1, 2, 52, 65], [1, 2, 58, 65], [1, 2, 59, 65], [1, 2, 60, 65], [1, 2, 61, 65], [1, 2, 63, 65], [1, 6, 1, 65], [1, 6, 12, 65], [1, 6, 21, 65], [1, 6, 22, 65], [1, 6, 27, 65], [1, 6, 41, 65], [1, 6, 47, 65], [1, 6, 52, 65], [1, 6, 55, 65], [1, 6, 56, 65], [1, 6, 61, 65], [1, 6, 64, 65], [1, 7, 1, 65], [1, 7, 9, 65], [1, 7, 12, 65], [1, 7, 21, 65], [1, 7, 22, 65], [1, 7, 24, 65], [1, 7, 27, 65], [1, 7, 29, 65], [1, 7, 30, 65], [1, 7, 33, 65], [1, 7, 41, 65], [1, 7, 44, 65], [1, 7, 47, 65], [1, 7, 52, 65], [1, 7, 55, 65], [1, 7, 59, 65], [1, 7, 61, 65], [1, 7, 64, 65], [1, 10, 1, 65], [1, 10, 9, 65], [1, 10, 12, 65], [1, 10, 22, 65], [1, 10, 24, 65], [1, 10, 25, 65], [1, 10, 29, 65], [1, 10, 30, 65], [1, 10, 33, 65], [1, 10, 38, 65], [1, 10, 41, 65], [1, 10, 43, 65], [1, 10, 44, 65], [1, 10, 47, 65], [1, 10, 58, 65], [1, 10, 63, 65], [1, 11, 1, 65], [1, 11, 9, 65], [1, 11, 21, 65], [1, 11, 22, 65], [1, 11, 30, 65], [1, 11, 41, 65], [1, 11, 44, 65], [1, 11, 47, 65], [1, 11, 49, 65], [1, 11, 50, 65], [1, 11, 52, 65], [1, 11, 56, 65], [1, 11, 58, 65], [1, 11, 60, 65], [1, 17, 1, 65], [1, 17, 2, 65], [1, 17, 9, 65], [1, 17, 12, 65], [1, 17, 21, 65], [1, 17, 24, 65], [1, 17, 25, 65], [1, 17, 27, 65], [1, 17, 30, 65], [1, 17, 33, 65], [1, 17, 38, 65], [1, 17, 44, 65], [1, 17, 47, 65], [1, 17, 50, 65], [1, 17, 58, 65], [1, 17, 59, 65], [1, 17, 60, 65], [1, 17, 64, 65], [1, 18, 1, 65], [1, 18, 2, 65], [1, 18, 3, 65], [1, 18, 9, 65], [1, 18, 24, 65], [1, 18, 27, 65], [1, 18, 29, 65], [1, 18, 41, 65], [1, 18, 44, 65], [1, 18, 47, 65], [1, 18, 49, 65], [1, 18, 55, 65], [1, 18, 56, 65], [1, 18, 58, 65], [1, 18, 60, 65], [1, 18, 61, 65], [1, 18, 64, 65], [1, 21, 1, 65], [1, 21, 9, 65], [1, 21, 12, 65], [1, 21, 22, 65], [1, 21, 24, 65], [1, 21, 25, 65], [1, 21, 27, 65], [1, 21, 30, 65], [1, 21, 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65], [1, 25, 63, 65], [1, 25, 64, 65], [1, 27, 1, 65], [1, 27, 2, 65], [1, 27, 9, 65], [1, 27, 21, 65], [1, 27, 30, 65], [1, 27, 38, 65], [1, 27, 47, 65], [1, 27, 49, 65], [1, 27, 56, 65], [1, 27, 58, 65], [1, 27, 60, 65], [1, 27, 64, 65], [1, 28, 1, 65], [1, 28, 3, 65], [1, 28, 9, 65], [1, 28, 21, 65], [1, 28, 25, 65], [1, 28, 29, 65], [1, 28, 30, 65], [1, 28, 50, 65], [1, 28, 52, 65], [1, 28, 58, 65], [1, 28, 60, 65], [1, 28, 63, 65], [1, 29, 1, 65], [1, 29, 2, 65], [1, 29, 3, 65], [1, 29, 9, 65], [1, 29, 24, 65], [1, 29, 25, 65], [1, 29, 30, 65], [1, 29, 38, 65], [1, 29, 44, 65], [1, 29, 47, 65], [1, 29, 49, 65], [1, 29, 55, 65], [1, 29, 56, 65], [1, 29, 58, 65], [1, 29, 60, 65], [1, 29, 61, 65], [1, 29, 63, 65], [1, 29, 64, 65], [1, 30, 1, 65], [1, 30, 12, 65], [1, 30, 21, 65], [1, 30, 24, 65], [1, 30, 27, 65], [1, 30, 29, 65], [1, 30, 33, 65], [1, 30, 41, 65], [1, 30, 43, 65], [1, 30, 50, 65], [1, 30, 52, 65], [1, 30, 55, 65], [1, 30, 56, 65], [1, 30, 58, 65], [1, 30, 61, 65], [1, 30, 64, 65], [1, 32, 1, 65], [1, 32, 9, 65], [1, 32, 21, 65], [1, 32, 29, 65], [1, 32, 38, 65], [1, 32, 47, 65], [1, 32, 49, 65], [1, 32, 52, 65], [1, 32, 55, 65], [1, 32, 58, 65], [1, 32, 59, 65], [1, 32, 63, 65], [1, 32, 64, 65], [1, 34, 1, 65], [1, 34, 9, 65], [1, 34, 22, 65], [1, 34, 43, 65], [1, 34, 47, 65], [1, 34, 50, 65], [1, 34, 58, 65], [1, 34, 61, 65], [1, 34, 63, 65], [1, 34, 64, 65], [1, 35, 1, 65], [1, 35, 2, 65], [1, 35, 9, 65], [1, 35, 12, 65], [1, 35, 21, 65], [1, 35, 29, 65], [1, 35, 38, 65], [1, 35, 41, 65], [1, 35, 43, 65], [1, 35, 44, 65], [1, 35, 47, 65], [1, 35, 49, 65], [1, 35, 50, 65], [1, 35, 56, 65], [1, 35, 58, 65], [1, 35, 59, 65], [1, 35, 61, 65], [1, 36, 1, 65], [1, 36, 2, 65], [1, 36, 3, 65], [1, 36, 9, 65], [1, 36, 21, 65], [1, 36, 27, 65], [1, 36, 29, 65], [1, 36, 41, 65], [1, 36, 44, 65], [1, 36, 47, 65], [1, 36, 52, 65], [1, 36, 60, 65], [1, 36, 61, 65], [1, 39, 1, 65], [1, 39, 3, 65], [1, 39, 9, 65], [1, 39, 21, 65], [1, 39, 22, 65], [1, 39, 24, 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[1, 46, 22, 65], [1, 46, 25, 65], [1, 46, 27, 65], [1, 46, 29, 65], [1, 46, 30, 65], [1, 46, 43, 65], [1, 46, 47, 65], [1, 46, 50, 65], [1, 46, 52, 65], [1, 46, 58, 65], [1, 46, 60, 65], [1, 46, 61, 65], [1, 46, 63, 65], [1, 46, 64, 65], [1, 47, 1, 65], [1, 47, 2, 65], [1, 47, 9, 65], [1, 47, 22, 65], [1, 47, 27, 65], [1, 47, 29, 65], [1, 47, 38, 65], [1, 47, 41, 65], [1, 47, 44, 65], [1, 47, 49, 65], [1, 47, 50, 65], [1, 47, 55, 65], [1, 47, 56, 65], [1, 47, 59, 65], [1, 47, 63, 65], [1, 47, 64, 65], [1, 48, 1, 65], [1, 48, 9, 65], [1, 48, 12, 65], [1, 48, 22, 65], [1, 48, 25, 65], [1, 48, 29, 65], [1, 48, 30, 65], [1, 48, 38, 65], [1, 48, 43, 65], [1, 48, 47, 65], [1, 48, 52, 65], [1, 48, 55, 65], [1, 48, 61, 65], [1, 48, 63, 65], [1, 50, 1, 65], [1, 50, 2, 65], [1, 50, 3, 65], [1, 50, 12, 65], [1, 50, 21, 65], [1, 50, 22, 65], [1, 50, 24, 65], [1, 50, 25, 65], [1, 50, 30, 65], [1, 50, 33, 65], [1, 50, 38, 65], [1, 50, 43, 65], [1, 50, 44, 65], [1, 50, 47, 65], [1, 50, 55, 65], [1, 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5. Look at the cities that border Piscataway
and for each of them those that border them
and again until you get to princeton

Piscataway (1)
Edison (2)
Franklin (3)
New Brunswick (4)
North Brunswick (5)
South Brunswick (6)

(i) construct the graph
[{2, 3, 4, 6}, {1, 4, 6}, {1, 5}, {1, 2}, {3}, {1, 2}]

(ii) find the number of paths from piscataway to princeton
58
(a) find the actual set using paths (and list them)
(b) by using NuPaths
(c) by using GFt
[-(t - 1)*(t + 1)^2*t/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1), -t*(t^3 + 2*t^2 - t - 1)/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1), t^2*(t + 1)/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1), -t^2*(2*t^3 + 3*t^2 - 2*t - 2)/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1), t^3*(t + 1)/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1), (t - 1)*(t + 1)*(2*t^3 + 4*t^2 - 1)/(4*t^5 + 7*t^4 - 4*t^3 - 7*t^2 + 1)]