Probability problem - how did Princeton Review solve this?

maxim730

I can't figure out how Princeton Review solved the below problem, their explanation doesn't make sense to me.

If anyone can break it down, that would be great!!

Q:

Ten strips of paper are numbered 1 to 10 and placed in a bag. If three numbers are drawn from the bag at random, what is the probability that the sum of the numbers drawn will be odd?

A) 1/12
B) 5/36
C) 15/36
D) 1/2
E) 11/18

OA is D

Princeton explanation (This is an image I added, I didn't want to type it out_):

http://img143.imageshack.us/my.php?image=problemzn4.jpg

Mark Dabral

Hi maxim730,

The total number of different ways one can select a distinct set of three numbers from the set of 10 number is given by:

10C3 = 10!/(10-3)!3! = 7x8x9/(3x2x1) = 120

There only two different ways that the sum of three integers can be an odd number:

Case 1: Odd + Odd + Odd = Odd (All three numbers being odd)

So how many distinct sets of three odd numbers can be selected from 5 odd numbers. This is given by 5C3 = 5!/(5-3)!3! = 10

Case 2: Even + Even + Odd = Odd

Out of the 5 odd number we can select an odd number in 5 different ways. Similarly, we can select a set of two even numbers from a set of five even number in 5C2 ways or 5!/(5-2)!2! = 10

Therefore, the distinct sets of two even and one odd numbers are
5 x 10 = 50

Therefore, the total number of distinct sets of three numbers that add up to an odd number is 60. The probability is then given by the number of favorable events divided by the total number of possible outcomes which is 120 in this case, and is given by 60/120 = 1/2.

Let me know if any of the steps are not clear.

Cheers,
Mark

maxim730 · Posted: Thu Dec 28, 2006 7:49 am Post subject:

That makes sense, thanks Mark!

thankont · Posted: Fri Dec 29, 2006 11:52 pm Post subject:

Mark can we just simply say that since there are same number of odd and even numbers from 1 - 10 (5 each), and since odd number can be formed the same number of ways as an even number then right away p=1/2 ?