ASSIGNMENT 1 WRITTEN Solution

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



if there are no restrictions?
if the langauages should alternate?



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



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




For a real number and a positive integer, show that



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



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



≥0, 1≤ ≤4



0, 1≤ ≤4



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.



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.