I have a scenario like this, from a deck of 40 cards, I draw 5 cards, and I need to know the probability of drawing at least 1, at most 3 for each card A, B and C, and each has 3 copies in the deck.
I came up with an expression like this:
$$ {{3 \choose 1}{3 \choose 1}{3 \choose 1}{31 \choose 2} \over {40 \choose 5}} + {{3 \choose 1}{3 \choose 1}{3 \choose 2}{31 \choose 1} \over {40 \choose 5}} + {{3 \choose 1}{3 \choose 2}{3 \choose 1}{31 \choose 1} \over {40 \choose 5}} + {{3 \choose 1}{3 \choose 2}{3 \choose 2} \over {40 \choose 5}} + {{3 \choose 2}{3 \choose 1}{3 \choose 1}{31 \choose 1} \over {40 \choose 5}} + {{3 \choose 2}{3 \choose 1}{3 \choose 2} \over {40 \choose 5}} + {{3 \choose 2}{3 \choose 2}{3 \choose 1} \over {40 \choose 5}} + {{3 \choose 1}{3 \choose 1}{3 \choose 3} \over {40 \choose 5}} + {{3 \choose 1}{3 \choose 3}{3 \choose 1} \over {40 \choose 5}} + {{3 \choose 3}{3 \choose 1}{3 \choose 1} \over {40 \choose 5}} $$
Is there any math expression I can use to simplify this expression? I tried to use summation, but it doesn't seems to be the best idea, as my expression would introduce double counting.