Kaplan is right on this one. Here is the solution:
[spoiler]The sum of all five numbers is 5*24=120
The sum of the numbers who have the average 18 is 18*2=36
Hence, the sum of other three numbers is 120-36=84.
Now, since we want the largest possible value, then the rest of the numbers should have the lowest value allowed, which is 8. From here you can see that you should make an equation that looks like this 8+8+x=84, where x is what you're looking for. x=>68. C[/spoiler]

Hey GMATJeet,

Great question - keep in mind that your goal here is crucial. You want to maximize the value of one number. And since you're fixed at an average of 24, your goal is then to minimize the value of the other numbers so that all the "disposable" value goes to the one largest number.

To minimize the value of the other four values, let's look at our constraints:

-the minimum value of any one value is 8
-two of the four need to average out to 18

So, strategically, we want as many 8s - the lowest possible number - as possible, so let's call two of the values 8. If the values are a, b, c, d, and e, let's call a=8 and b=8.

Then we're stuck with two values averaging out to 18, so the best we can do on c+d = 36.

So we're left with:

(8 + 8 + 36 + e)/5 = 24
16 + 36 + e = 120
52 + e = 120
e = 68


Again, the big key is the strategy - to maximize one value here your goal is then to minimize the others, so you have to figure out what allows you to do that.