#ATTENDANCE QUIZ FOR LECTURE 2 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p2
#with an attachment called
#p2FirstLast.txt
#(e.g. p2DoronZeilberger.txt)

#Right after finishing watching the Second lecture but no later than Friday, Sept. 11, 2020, 8:00pm\


#Q1. THE FIRST ATTENDANCE QUESTION WAS:
#
#Let a := FIRST DIGIT of your RUID
#Let b := SECOND DIGIT of your RUID
#Let c := FIFTH DIGIT of your RUID
#Let d := SIXTH DIGIT of your RUID
#How  many walks in Manhattan are there (not using Broadway) from the corner of min(a,b) St and
#min(c,d) Ave to max(a,b) St and max(c,d) Ave, only walking in the "positive" direction?

#A1. MY ANSWER TO THE FIRST ATTENDANCE QUESTION IS: 2002 Walks.
#
#NuWalks([1,0], [6,9])? /! Use Maple to solve. Thanks computer! !/
#NuWalks(m,n): inputs two non-negative integers m and n
#and outputs the SET of all POSITIVE walks {E,N} from [sx,sy] to [dx,dy]
#all the rearramgemets of E^nN^m

NuWalks:=proc(sx,sy,dx,dy) local W1,W2,w1,w2:
option remember:
if dx<sx or dy<sy then
 RETURN(0):
fi:
if sx=dx and sy=dy then
 RETURN(1):
fi:
W1:=NuWalks(sx,sy,dx-1,dy):
W2:=NuWalks(sx,sy,dx,dy-1):
W1+W2:
end:

RUID := [1,6,9,0,0,9,1,1,1]:
a := RUID[1]:
b := RUID[2]:
c := RUID[5]: 
d := RUID[6]:
ANS1 := NuWalks(min(a,b), min(c,d), max(a,b), max(c,d)): 
print("ANS1 - ", ANS1);
#Q2. THE  SECOND ATTENDANCE QUESTION WAS:
#
#Consider your RUID as a word of length 9, from the 10-letter alphabet {0,1,2,...,9}.
#How many ways to rearrange your RUID?

#A2. MY ANSWER TO THE  SECOND ATTENDANCE QUESTION IS: 3780 Permutations
#
# This is permutations with repititions allowed.
# Will use the theorem to solve first.
n := nops(RUID):
ANS2_1 := n!/(4!*2!*2!):

# Or we can use the "combinat" Maple Package...
with(combinat):
ANS2_2 := nops(permute(RUID)):
print("ANS2_1 - ",ANS2_1);
print("ANS2_2 - ", ANS2_2);