Hi, please, tell me if my was of solving this problem is good or not : Users of public transport are waiting for the bus: 3 women and 1 man. One of them wears glasses, another gloves, and another a hat. What is the probability that the man has none of these accessories?

Here how :

1:Woman
2:Woman
3:Woman
4:Man
                                               Glasses, Gloves, Hat, Nothing

All possible combinations :

{1,2,3,4} {1,2,4,3} {1,3,2,4} {1,3,4,2} {1,4,2,3} {1,4,3,2} {2,1,3,4} {2,1,4,3} {2,3,1,4} {2,3,4,1} {2,4,1,3} {2,4,3,1} {3,1,2,4} {3,1,4,2} {3,2,1,4} {3,2,4,1} {3,4,1,2} {3,4,2,1} {4,1,2,3} {4,1,3,2} {4,2,1,3} {4,2,3,1} {4,3,1,2} {4,3,2,1}

Probability of the man having nothing : 6/24 or 0,25

___________________________________________________

Now here's what the ''real'' answer is : 54/72 or 0,75

I figured that they could have obtained this answer by multiplying 18/24 X3 = 54/72 (In my opinion, this is an error because the question is what is the probability that the man has NONE of these accesories and 18/24 is the probability that he wears one of these objects! Am I correct?) So the ''real'' answer would be 18/72

And last thing that bugs me, if I'm correct in all my previous statements, why is the answer on 72 ? There is 24 possibilities ! Why multiply by 3 ? Does the order of the object change anything ? (WHich isn't logical, if you ask me)

Thank you