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.
