2005 AIME I Problem 5

Handle orientations and colors separately. Record each coin’s orientation, bottom to top, as $U$ (face up) or $D$ (face down).

Face to face happens exactly when a $U$ sits directly below a $D,$ so all the $D$’s must come before all the $U$’s

Choose the coin orientations and the gold/silver positions independently. Record the orientations from bottom to top as a string of $U$ (engraved face up) and $D$ (engraved face down). Two adjacent coins are face to face exactly when the lower coin’s engraved side faces up and the upper coin’s engraved side faces down — that is, exactly when a $U$ is immediately followed by a $D.$

A string of $U$’s and $D$’s avoids the pattern $UD$ exactly when every $D$ precedes every $U,$ so the string is $\text{D}^i\text{U}^{8-i}$ for some $i = 0, 1, \ldots, 8:$ there are $9$ allowable orientation strings. Independently, the gold coins occupy $4$ of the $8$ positions in $\binom{8}{4} = 70$ ways.

The total is $9 \cdot 70 = 630.$