This is a common question I've seen asked but I've searched a little and I'm not sure I've found what I'm looking for.
If I have a bag of 15 marbles where 7 are red/3 Blue and 5 green. If I pick 3 at random what is the probability that I get a marble of each colour? (I.e 1 red/green/blue).
Is there a difference between picking 1 at a time and picking all three at a time? Or can picking all three be thought as picking 1 at a time but 3 times? The total number of marbles decrease in the bag after you pick them so if you were to draw a tree diagram and try to find the probability out that way it would take a long time and it's inefficient
How does this quick way work ?
$\frac{7}{15}$ x $\frac{3}{14}$ x $\frac{5}{13}$ x $3!$
Or any of the numerators re arranged but the denominator kept the same. I do not understand how the anwser can be achieved like this but I understand that there are 6 possible routes which is what 3 factorial is for. Why do you multiply them together even though there are different ways? I.e When you pick there are different choices like this:
$\frac{7}{15}$ , $\frac{5}{14}$ , $\frac{3}{13}$
$\frac{7}{15}$ , $\frac{3}{14}$ , $\frac{5}{13}$
$\frac{5}{15}$ , $\frac{3}{14}$ , $\frac{7}{13}$
$\frac{5}{15}$ , $\frac{7}{14}$ , $\frac{3}{13}$
$\frac{3}{15}$ , $\frac{5}{14}$ , $\frac{7}{13}$
$\frac{3}{15}$ , $\frac{5}{14}$ , $\frac{7}{13}$
How is the quicker way obtained ? I.e what is the method behind it.