Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, five white ones, and two 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 has all the red ones.
She has all the red ones.
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There are 15 marbles in the bag, so there are
15C8 = 15! / (8! 7!)
= 15*14*13*12*11*10*9 / (7*6*5*4*3*2*1)
= 13*11*5*9 = 6,435 different combinations of 8 of them.
The number of different combinations including all 4 red ones is the number of different ways 4 of the other 11 marbles can be combined:
11C4 = 11! / (7! 4!)
= 11*10*9*8 / (4*3*2*1)
= 11*10*3 = 330
So the probability is
330/6,435 = 2/39
15C8 = 15! / (8! 7!)
= 15*14*13*12*11*10*9 / (7*6*5*4*3*2*1)
= 13*11*5*9 = 6,435 different combinations of 8 of them.
The number of different combinations including all 4 red ones is the number of different ways 4 of the other 11 marbles can be combined:
11C4 = 11! / (7! 4!)
= 11*10*9*8 / (4*3*2*1)
= 11*10*3 = 330
So the probability is
330/6,435 = 2/39