Question 905735: A quality control inspector samples 6 computer chips from a box containing 11 computer chips. If the box contains 2 defective chips, in how many ways can he select a sample so that all 6 are good?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe your answer will be c(9,6) = (9*8*7*6*5*4) / 6*5*4*3*2*1) = (9*8*7) / (3*2*1) = (3*4*7) = (12*7) = 84
you have 11 to choose from but only 9 are good because 2 are bad.
the total possible ways you can get a sample is c(11,6) but the total possible ways you can get a sample where all are good is c(9,6).
i looked at this with much smaller numbers to see if i could come up with the formula.
it was confirmed through several tries, the most complex being the one shown below:
assume a sample of 5 computer chips and 2 are defective.
you select a sample of 3.
in how many ways can you get all good?
the total number of possible ways to get any sample is c(5,3) = (5*4*3) / (3*2*1) = (5*4) / (2*1) = 10
the total number of possible ways to get all good is c(3,3) = 1.
if the formula is correct, there should only be one out of the 10 possible ways to get all three good.
assume the 5 chips are chip a,b,c,d,e
the total possible combinations of 3 are shown below. there should be 10 of them.
assume d and e are defective.
we'll mark the number of sets that contain either d or e in them with asterisks
abc
abd ***
abe ***
acd ***
ace ***
ade ***
bcd ***
bce ***
bde ***
cde ***
the only set that didn't contain d or e in it was abc which is only 1 out of the 10 possible sets.
the formula was 3C3 instead of 5C3 because 2 were defective and we only wanted all good.
i applied the same logic to your problem, so instead of 11c6, you drew 9c6 which gave you all the possible combinations where all were good.
note that 9c6 and c(9,6) are two different ways to represent the same thing, namely how many different ways to draw sets of 6 from 9 where each set is different from the other and where order doesn't matter.
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