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).

Discrete Structures - Solving Systems of Equations

Solve the following systems of equations: (a) x=4 (5) and x=7 (11) (b) 3=34 (100) and x=-1 (51) *Please see attachment for proper symbols and complete instructions

Factor Positive Integrers into Primes

Factor into primes the following positive integers: (a) 25 (b) 4200 (c) 10(to the exponent)10 (d) 19 (e) 1 *Please see attachment for proper citation and complete instructions

Lowest Common Multiple (Prime Factorizations)

Let a and b be integers. A common multiple of a and b is an integer n for which a|n and b|n. We call an integer m the least common multiple of n provided (1) m is positive, (2) m is a common multiple of a and b, and (3) if n is any other positive common multiple of a and b, then n [greater than or equal to] m. The notation fo ...continues

Perfect Integers

An integer 'n' is called 'perfect' if it equals the sum of all its divisors 'd' ... {see attachment for complete definition and example} Let 'a' be a positive integer. Prove ... {see attachment}

Perfect Square; Perfect Cube; Perfect Fifth Power

Find the smallest positive integer N such that N/2 is a perfect square, N/3 is a perfect cube and N/5 is a perfect fifth power.