SOLUTION: Problem: A bag contains five real diamonds and seven fake diamonds. If seven diamonds are picked from the bag at random, what is the probability that exactly three of them are real

Question 1070003: Problem: A bag contains five real diamonds and seven fake diamonds. If seven diamonds are picked from the bag at random, what is the probability that exactly three of them are real?
My Attempt:
I found how many ways we can pick 7 from 12 using the combination formula, 792.
I wasn't sure about this step: I found how many ways I can pick 3 real diamonds from the 5, which was 10.
I never found a answer as a wasn't sure where to go now... I do know the correct answer is 175/396.
Did I go wrong somewhere?If not, what do I do now? Thanks in Advance!
Answer by swincher4391(1107) (Show Source): You can put this solution on YOUR website!
So the idea is that we want to find the total number of ways that we can pick 7 diamonds (real or fake) from the 12 diamonds.
So as you said, 12 choose 7 = 792
Now we need the total number of ways that we can satisfy that of the 7 we picked, exactly three are real, but this also means that four are fake.
So to choose 3 real from 5 is 5 choose 3 = 10
And to choose 4 fake from 7 is 7 choose 4 = 35
= =

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