1. ## probability

Four bottles are filled with grade A sauce and three with grade B sauce . If three bottles are randomly selected from the seven bottles , what is the probability that both grades of sauces are in teh selection ?

I know this can be done by drawing the tree diagram , then sum all the probabilities . I just want to know whether this can be done by combination .

something / 7C3

i just cant figure out that something .

2. Hello, thereddevils!

Four bottles are filled with grade A sauce and three with grade B sauce.
If 3 bottles are randomly selected from the seven bottles,
what is the probability that both grades of sauces are in the selection?
There are: . ${7\choose3} = 35$ possible samples.

There are two possible scenarios:

. . (1) Two A's and one B

. . . . $P(2A,1B) \:=\:\frac{{4\choose2}{3\choose1}}{35} \:=\:\frac{18}{35}$

. . (2) One A and two B's

. . . . $P(1A,2B) \;=\;\frac{{4\choose1}{3\choose2}}{35} \:=\:\frac{12}{35}$

Therefore: . $P(\text{one of each}) \;=\;\frac{18}{35} + \frac{12}{35} \;=\;\frac{30}{35} \;=\;\frac{6}{7}$

Someone else can show you the "complement" approach.
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