Unity作业|代写Assignment – 这是一个典型的unity3D题目代写
University of Toronto at Scarborough
CSCA67/MATA67Discrete Mathematics, Fall 2018
assignment #2: Counting / Probability
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This assignment is due by 11:59 p.m. December 3. If you havent finished by then, you may hand in your assignment late with a penalty as specified in the course information sheet.
1.[10] Recall the proof thatC(n,3)triangles are formed bynnon-parallel lines in the plane if no three of
the lines intersect at a single point. Repeat this proof withnlines total,mof whichareparallel.
2.[5] The following two claims are similar to the claim in the triangle problem discussed in lecture, but
there are subtle differences. Either prove or disprove each claim.
(a) LetT(n)be:C(n,3)triangles are formed bynlines in the plane if no three of the lines intersect
at a single point.nN,n 3 ,T(n).
(b) LetR(n)be:C(n,3)triangles are formed bynnon-parallel lines in the plane.nN,n 3 ,
R(n).
3.[10] Draw Pascals triangle as shown in lecture. Label each row from the top starting at 0 , and each entry
in a row starting at 0 from the left edge. Using either simple or strong induction, prove that the entries
in then-th row sum to 2 n.
4.[10] How many integer solutions are there to the equation
x 1 +x 2 +x 3 + 2x 4 +x 5 = 72
wherex 1 2 ,x 2 , x 3 1 , andx 4 , x 5 0.
5.[10] Consider the problem of arranging 5 As, 6 Bs and 3 Cs.
(a) How many arrangements are there? Count in twodifferent ways... first using permutation with
repetition, and then using selection.
(b) Repeat (a) with the added requirement that all Bs must be in groups of 2 , 4 or 6. For example,
ABBCABBABBCAAC and BBBBAABBAAACCC are valid sequences but BBBABBBCCAA-
CAA is not.
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(c) Repeat (a) with the added requirement that the first A precedes the first B which precedes the
first C. For example, AABABCBBAACBCB is a valid sequence but AAACBAACBCBBBB and
CAAABAACBCBBBB are not.
6.[10] An urn contains seven red balls, seven white balls, and seven blue balls. A sample of five balls is
drawn at random without replacement. Compute the probability that:
(a) The sample contains four balls of one colour and one ball of another colour.
(b) All of the balls are of the same colour.
(c) The sample contains at least one ball of each colour.
7.[10] A game show offers contestants the following chance to win a car: There are three doors. A car is
hidden behind one door, and goats are hidden behind each of the other doors. The contestant selects
a door. The game show host then opens one of the doors not chosen to reveal a goat (there are two
goats, so there is always such a door to open). At this point, the contestant is given the opport unity to
stand pat (do nothing) or to choose the remaining door. Suppose you are the contestant, and suppose
you prefer the expensive sports car over a not-so-expensive goat as your prize. What do you do?
(a) Suppose you decide to stand with your original choice. What are your chances of winning the
car?
(b) Suppose you decide to switch to the remaining door. What are your chances of winning the car?
(c) Suppose you decide to flip a fair coin. If it comes up heads, you change your choice, otherwise,
you stand pat. What are your chances of winning the car?
8.[10] Suppose there is a noisy communication channel in which either a 0 or a 1 is sent with the following
probabilities:
- probability a 0 is sent is 0.
- probability a 1 is sent is 0.
- probability that due to noise, a 0 is changed to a 1 during transmission is 0.
- probability that due to noise, a 1 is changed to a 0 during transmission is 0.
Suppose that a 1 is received. What is the probability that a 1 was sent? You must use bothBayes
Ruleand the concept ofTotal Probabilityto answer this question. Show all of your work.
[total: 75 marks]
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