Find the probability of each event. Answer IN FRACTION form.

A bag contains five real diamonds and five fake diamonds. If five diamonds are picked from the bag at random, what is the probability that all of them are real?

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

3x10/792= 350/792= 175/396

let x and 400 - x be the two numbers.  

==> 0.80x + 0.85(400 - x) = 400 - 68 from the statement of the problem.  

< ==> 0.80x + 0.85(400 - x) = 400 - 68  

==> 340 - 0.05x = 332  

< ==> -0.05x = -8 ==> x = 160, the first number ==> 400 - x = 240, the second number.