MGMAT Advanced Quant, Workout Sets 7, Q67

[budoshi]

Set S is the set of all prime integers between 0 and 20. If the three numbers are chosen randomly from set S, what is the probablity that the sum of these three numbers is odd?

Answer is 5/8. The solution explains there are 8 possible prime integers and for the sum to be odd, the choices will be Odd+Odd+Odd. Hence, 7/8*6/7*5/6 = 5/8. I get it...HOWEVER, there were 2 other methods that I am very confused about and was wondering why these lead to the WRONG answer:

1st method:
1-Probability that the sum is Even. Using the same methodology as above, this would be: 1- (1/8*7/7*6/6)=7/8. - Wrong answer.

2nd method - Counting
How would you do this using the counting method i.e. 8C3=56 I get stuck with trying to figure out the number of ways to sum to odd.

Thank you

[tim]

1 - this will totally work if you calculate it correctly. you used incorrect calculations, which is why you got the wrong answer.

2 - if you figured out the total number of ways is 8C3, it should follow naturally that the number of ways to get three odds is 7C3, because 7 of the numbers are odd.