Matching colour counts in two pairs of jellybeans

A bag holds $6$ green and $10$ yellow jellybeans. Four are taken out one after another at random without replacement. What is the probability that the first two drawn and the last two drawn contain the same colours as multisets (order within each pair ignored)? For example, GYYG and GGGG qualify while GGYG does not.

$\tfrac{135}{364}$. Either all four beans share a colour, or each pair holds one green and one yellow (in $2\times2=4$ possible orders). Ordered draws give $360$ all-green and $5040$ all-yellow sequences plus $4\cdot2700=10800$ mixed ones, out of $16\cdot15\cdot14\cdot13=43680$, which simplifies to $\tfrac{135}{364}$.