I have been struggling with this math problem can you solve it?

Question

(I made this problem up myself because this is how I put away my own cups)

There is an empty cabinet that needs to get filled with cups. It can fit three rows of five with two cups high. With thirty gray and thirty yellow cups at your disposal, how many possible combinations of cups (that don’t use more than thirty cups total) are there that don’t fulfill the pattern? Show your work.

Pattern Rules

1. The middle spot in each row must be empty or gray*
2. The other spots must be empty or yellow*
3. Each row across must be symmetrical by height (0 cups, 1 cup, 2 cups)

*Only the completely visible cup (One cup high then that cup, otherwise just the top cup) counts

Example of a working pattern...

Answer 1

Let A be the top cup and B be the bottom cup.

There are 3 rows and 5 columns. will use (row,column,A or B) notation.

So

(1,3, A)

(2,3, A)

(3,3, A)

MUST be Gray

27 gray and 30 yellow cups remain, for a total of 57 available cups.

There are 27 positions to fill.

(57 choose 27) =

57*56*55* .... * 28 / (30*29*28*27 ....)

= 14031391033119152