How many ways to cover this rectangle

[sharpycasio]

Homework Statement

In how many different non-overlapping ways can a (2 x 10) rectangle be covered by (1 x 1) and (1 x 3) rectangles.

Homework Equations

The Attempt at a Solution

I've never done any question like this. The only solution I can think of is a brute force method but I doubt that's the way it is intended to be solved. Any hints? Thanks.

 

[Mark44]

Homework Statement

In how many different non-overlapping ways can a (2 x 10) rectangle be covered by (1 x 1) and (1 x 3) rectangles.

Homework Equations

The Attempt at a Solution

I've never done any question like this. The only solution I can think of is a brute force method but I doubt that's the way it is intended to be solved. Any hints? Thanks.

Since the smaller rectangles can't overlap the larger (2 x 10) rectangle, you can simplify matters by looking at how you would cover a 1 x 10 rectangle with 1 x 1 squares and 1 x 3 rectangles.

 

[sharpycasio]

Since the smaller rectangles can't overlap the larger (2 x 10) rectangle, you can simplify matters by looking at how you would cover a 1 x 10 rectangle with 1 x 1 squares and 1 x 3 rectangles.

That's true! Thanks. So if I did it correctly this what I got for a 1 x 10 rectangle. (For simplicity, rectangle refers to 1x3 and square refers to 1x1)0 rectangles and 10 squares: 1
1 rectangle and 7 squares: 8
2 rectangle and 4 squares: 15
3 rectangle and 1 squares: 4Total = 28 ways to cover a 1 x 10 rectangle.

How would I extrapolate from this to a 2 x 10 rectangle? Would it be 28²? Thanks.

 

[Mark44]

I haven't checked your figures, but for the 2 x 10 rectangle, 27² seems reasonable. For each of the 27 ways on one of the 1 x 10 strips, there are 27 different arrangements on the other strip.

 

[sharpycasio]

I initially made a mistake (missed one arrangement). Right now I have 28. Thanks. (Previous post has been edited)