Mathematics For IT project 6


Semester 2, 2018



Discrete Maths

1. Consider the sequence defined recursively by

s0 = 0, s1 = 1, s2 = 0; sn := sn−3 + sn−2 − sn−1for 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?