Suzan sees a bag of marbles, she grabs a handful at random. she has seen a bag containing four red marbles, two green ones, four white ones, and three purple ones. she grabs eight of them. find the probability of the following event, expressing it as a fraction in lowest terms. hint [see example 1.] she does not have all the green ones.

In this question, Suzan is taking 8 marble from a bag containing 4 red marbles, 2 green ones, 4 white ones, and 3 purple ones. The total marble in the bag should be: 4+2+4+3= 13. The question is how much the probability to get <2  green marble. In this question, the red, white and purple are not specified so you can join them into "not-green group". It will be easier to find the probability of "have all the green one" which mean Suzan take 2 green marble and 6 not-green marble from 11 marble

Ways to take 8 marble out of 12=12C5= 12!/4!8!= 495
Ways to take 2 green marble out of 2= 2C2= 1
Ways to take 6 not-green marble out of 11= 11C6= 11!/6!5!= 462

Probability to have all the green ones= 1*462/495= 154/165
Then the probability does not have all the green ones= 165/165 - 154/165= 11/165