## Chessboard by dominoes.

1) Find the number of different perfect covers of a 3-by-4 chessboard by dominoes.( enumerate all possibilities by drawing them)
2) Determine all shortest routes from A to B in the system of intersections and streets (graph) in the following diagram. The numbers on the streets represent the lengths of the streets measured in terms of some unit.

3) (briefly explain in words how you arrived at your answer)For each of the four subsets of the two properties (a) and (b), count the number of four-digit numbers whose digits are either 1,2,3,4, or 5:
(a) The digits are distinct.
(b) The number is even.
Note that there are four problems here: Null 0(no further restriction), {a} (property (a) holds), {b} (property (b) holds), {a,b} (both properties (a) and (b) hold).

4) How many orderings are there for a deck of 52 cards if all the cards of the same suit are together?
5) How many distinct positive divisors does each of the following numbers have?
(a) 3^4 x 5^2 X 7^6 x 11
(b) 620
6) Explain your reasoning completely for full credit:
For the game “SET,” the whole deck contains 81 cards with the attributes discussed in one of our videos. From this deck, for any 2 cards drawn, explain why you can always find a unique third card that makes a set as defined in the game.
For your reference: Recall the rules of how to form a ‘set’ in the game SET discussed in one of our videos : you have to find a set of 3 cards such that for each of the 4 attributes of shape, color, number of shapes and shading, the 3 cards you claim are a ‘set’ must either all be alike with respect to the attribute, or all be completely different with respect to the attribute.
Here are the attributes for your reference:
A) color (red, purple or green)
B) shape (oval, squiggle or diamond),
C) number (one, two or three)
D) shading (solid, striped or outlined)