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代写数学Mathematics For IT Assignment 6北美代写理论题

2018-09-24 08:00 星期一 所属: 其他代写 浏览:935

Mathematics For IT Assignment 6

 

Semester 2, 2018

 


Discrete Maths

 

1. Consider the sequence defined recursively by

s0 = 0, s1 = 1, s2 = 0; sn := sn3 + sn2  sn1 for n 3 .

(a) Calculate s3, s4  and s5  and make a guess as to the general form  of sn.

(b) Use induction to prove that your guess is correct.

2. For a positive integer n, let p(n) be the statement n2  3n is odd”.

(a) Show that p(n) p(n + 1) for each positive integer n.

(b) For which numbers n is p(n) true?

3. Let T (n) = T (n 1) + 2T (| 2 ) for n 1, let T (0) = 1, (where |·∫ is

the floor function).

(a) Evaluate T (1), T (2) and T (3).

(b) Use induction to prove that T (n) is divisible by 3 for all n  1.

 


Probabillity

 

4. (a) Determine P (9, 3) showing working.

(b) Two of the partitions of 9 into 3 parts can be given as 7 + 1 + 1 and 5 + 2 + 2. List all of the partitions and hence verify your answer from the first part.

5. Calculate A(6, 3). You must make use of the formulae given in lectures. Note that you may also quote any results which were already calculated in lectures, for example A(4, 2) = 14, if required.

6. In all of the following questions 7 coupons are being distributed into 4 showbags. In each case determine the number of ways it can be done, showing working.

(a) Suppose that the coupons and showbags are identical and no bag is empty.

(b) suppose that the coupons and showbags are identical but bags are allowed to be empty.

(c) Suppose that there are 4 different types of show bag and the coupons are identical and the bags are non-empty.


(d) Suppose the coupons and bags are both distinct but the bags are allowed to be empty.

7. In a very simple weather model we suppose that the chance of rain each day is 0.3, and is independent of the weather the previous day. In a month of 30 days, what is the probability of exactly 5 rainy days?

 

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