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?

Guest Jun 23, 2019

#1**+3 **

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

CPhill Jun 24, 2019

#3**+3 **

**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 = 2^{3} = 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:

hectictar Jun 24, 2019