1.
In how many patterns can six identical L-shaped pieces, each consisting of three unit squares, be placed to perfectly cover a fixed 3 by 6 board? One such pattern is shown.
2.
How many paths are there from C to D on the grid shown, if every step must be down or to the right?
+128460 2. We have the following moves (R R R R D D D D D D)
So in the set....we can choose any 4 of the 10 positions for the R's to occupy or, alternatively, any 6 of the 10 positions for the D's to occupy
So....the nuber of possible paths = 10C 4 = 10C6 = 210 possible paths
+9466 1. We will have to fit the pieces into 3 compartments like this:
If we try to place a piece between these compartments, it will create an unfillable area.
Within each of these compartments, there are only 2 ways to fit the pieces:
So the total number of possibilities = 2 * 2 * 2 = 23 = 8
They are:
1. AAA
2. AAB
3. ABA
4. ABB
5. BAA
6. BAB
7. BBA
8. BBB
And here is a picture of them:
