Multinomial Distribution Two-Up Game Probability of getting at least 1 of all possible outcomes

Question

For a Two-Up Game where two coins are tossed the 3 possible outcomes are $HH, HT, TT$
I'm trying to find the probability of getting at least one of each outcome when the game is played 5 times.

I don't think I'm doing this right as it turns out to be negative: $$P(Y_1>0, Y_2>0, Y_3>0)=1-P(Y_1=0, Y_2=0, Y_3=0)$$ $$= 1-(\frac{5!}{0!x0!x0!}\frac{1}{4}^0\frac{1}{4}^0\frac{1}{2}^0)$$ Can anyone help me with this? Thank you!

Answer 1

since you are finding the probability of getting at least one of each outcome, wouldnt you be trying to find P(Y1>1,Y2>1,Y3>1)?