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Here is a counting method I developed, which I have found especially useful. It is probably recorded in some books too.
It counts the number of ways to kill two birds with one stone, for instance enter and hit, enter and cover, hit and cover. It counts only moves played by two different checkers, e.g. it does not count as "enter and hit" a move that enters a checker and hits an indirect shot with the same checker.
Let n be the total number of dices that kill at least one bird. Let a be the number of dices that kill bird one, but not bird two. Let b be the number of dices that kill bird two, but not bird one. Then the number of rolls that kill both birds is
n^2 - a^2 - b^2(because it is the number of rolls that kill at least one bird, minus the rolls that kill only one of the birds).
E.g. if you enter with 1,2,4,5, and hit on the other side of the board with 2 or 3, then we have n = 5 (dices 1,2,3,4,5) ; a = 3 (1,4,5) ; b = 1 (3) ; so that you have 5^2 - 3^2 - 1^2 = 25-9-1 = 15 rolls that enter and hit.