Conditional Arrangement of objects in a row

Question

Arrange

$$a,a,a,b,b,b,c,c,c,c$$

in a row such that there is at least one $b$ and one $c$ between any two $a$.

Answer 1

We have $3$ 'a's, and imagine 4 buckets between them.

$()a()a()a()$

We know a 'b' and 'c' must go in each of the middle buckets, ands we have 'bcc' left over.

$()a(bc)a(bc)a()$

There are $[xy^2]:\left((1+x)(1+y+y^2)\right)^4=40$ (WA) ways to disperse the 'bcc's.

Then multiply by the number of arrangements of each bucket for each of the $40$ placings.

Explicitly, the options for a '3-bucket' are:

$$(1+x+y+xy+yx+yy+xyy+yxy+yyx)\\=(1+x+y+2xy+3xy^2)$$

And for the two middle '5-buckets':

$$(2xy+3x^2y+3xy^2+6x^2y^2+4xy^3+10x^2y^3)$$

So:

$$[x^3y^4]:(1+x+y+2xy+3xy^2)^2(2xy+3x^2y+3xy^2+6x^2y^2+4xy^3+10x^2y^3)^2$$

=358 (WA)