Computational Mathmatics

Victorian Institute of Technology Pty Ltd

ABN: 41 085 128 525 RTO No: 20829 TEQSA ID: PRV14007 CRICOS Provider Code: 02044E

ITSU2011 – Computational Mathematics Page 1 of 2

ITSU2011

Computational Mathematics

Activity 09

Victorian Institute of Technology Pty Ltd

ABN: 41 085 128 525 RTO No: 20829 TEQSA ID: PRV14007 CRICOS Provider Code: 02044E

ITSU2011 – Computational Mathematics Page 2 of 2

Part A)

1) If Mike has an infinite number of red, blue, yellow, and black socks in a drawer, how many socks must the Martian pull out of the drawer to guarantee he has a pair?

2) Suppose 𝑆 is a set of 𝑛 + 1 integers. Prove that there exist distinct 𝑎,𝑏 ∈ 𝑆 such that 𝑎 − 𝑏 is a multiple of 𝑛.

3) Show that in any group of 𝑛 people, there are two who have an identical number of friends within the group.

4) Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?

a. ½ b. 3/5 c. 2/3 d. 4/5 e. 1

5) Seven-line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.

6) Prove that having 100 whole numbers, one can choose 15 of them so that the difference of any two is divisible by 7.

Part B)

1) Expand

8

2 2

3  

  

 − x

x .

2) Find the coefficient of 3

x in the expansion of

5

3

2

1 4 



  

 +

x x .

3) What is the constant term in the expansion of

12

3

5 2 



  

 −

x x ?