Discrete Math : Permutations - N Objects in P Containers

Please see the attached file for the fully formatted problem. At the Clone Zone School a class of 10 identical boys and 8 identical girls go to a cafeteria. There are three registers : A, B and C. In how many ways can the students line up at the three registers?

Combinatorial Proof Based on the Binomial Formula

Please see the attached file for the fully formatted problem. 4. Provide a combinatorial proof: For postive integers N....

Find an explicit formula

5. Let TN be the number of ways a 2 X N rectangle can be tiled with 2 X 2 squares... (see attachment)

Counting

6. Eight people are attending a seminar in a room with eight chairs. In the middle of the seminar, there is a break and everyone leaves the room... (see attachment for rest of question)

Divisors and relative primes

Let a be an integer. Prove that 2a + 1 and a^2+ 1 are relatively prime. ( relative primes are numbers that their largest common divisor is 1).