Dice probabilities

[dm319]

So I came across this question while playing a new board game at the weekend.

This board game has multiple dice which are unusual - they have 2 blank faces, 2 'one dot' faces and 2 'two dot' faces. I.e. a 1/3rd chance to roll a zero, one or two.

In this game we roll multiple dice and then sum the number of dots to score that roll.

What is the probability of rolling a score of more than or equal to 7 when I have 5 of these dice in my hand?

Does anyone know of a good way to calculate this?

I've thought of two approaches, both of which are a bit inelegant. I'm curious to see if there is a neater way especially for the general case, and whether a program could be written to calculate it.