Solved-ASSIGNMENT 1 -WRITTEN -Solution

Description

1. I have seven different programming textbooks on my bookshelf, three C++ and four Java. In how many ways can I arrange the books

1. 1. if there are no restrictions?

   2. if the langauages should alternate?

1. 3. if all the C++ books must be next to each other?

1. 4. if all the C++ books must be next to each other and all the Java books must be next to each other?

2. a) Show that if is a positive integer and > 2, then

(₂)+( ⁻₂¹)

is a perfect square (i.e. its square root is an integer.)

1. 2. For a real number and a positive integer, show that

1. 2. 1. = (1+ ) −(₁) (1+ ) ⁻¹ +(₂) ²(1+ ) ⁻² −⋯+ (−1) ( )

3. Determine the number of integer solutions of ₁ + ₂ + ₃ + ₄ = 32, where

3. 1. ≥0, 1≤ ≤4

3. 2. >0, 1≤ ≤4

3. During the first six weeks after you graduate you send out at least one resumé a day but no more than 60 resumés in total. Show that there is a period of consecutive days during which you send out exactly 23 resumés.

3. Let ( , ₁) and ( , ₂) be two posets. Consider the set derived from the cross product of sets and

, × = {( , ): ∈ , ∈ }. Define relation  on × by (( , ), ( , )) ∈  if ( , ) ∈ ₁ and ( , ) ∈ ₂. Prove that  is a partial order.