How would I formulate the probability of this scenario?

Two players are on a team in a card game. They have separate decks that are largely unique but also have some duplicates of each others cards. Between them they want to draw a perfect series of cards for a combination.

Each deck has 15 cards

Player 1 has 3 of card A, 3 of card B, and 3 of card C.
Player 2 has 3 of card A, 3 of card B, 3 of card D and 3 of card E.

All other cards are undesired draws.

They want to draw 1 of card A, 1 of card B, 1 of card C, 1 of card D, and 1 more of either C, D, or E.

For the combination to work card A must be drawn on turn 1. Card B is the only card that may be drawn as late as turn 3. All other cards must be drawn on turn 1 or 2.

On turn 1 both players draw 2 cards from their deck to open the game. Thereafter they draw only 1 card each per turn.