i apologize for the screams of anguish that will result from doing this, but:
i can't draw a venn diagram on the page, and don't currently have the equipment to hand-draw one and scan it in. so:
click here for a reference venn diagram.
* the upper left circle (pdr1-3) will stand for set a.
* the upper right circle (pdr1*gad) will stand for set b.
* the bottom circle (pdre) will stand for set c.
--
(1) unless i'm reading this wrong, this statement is actually telling you the number you want. the 9 elements mentioned are in all three sets, and they're the only ones that are (think about this for a sec and it should sink in - if anything is in all three, then it's got to be one of those 16 in the first place, so you can be sure that you're not missing anything here). so that's definitely
sufficient.
(2) because of statement (1) we know that 9 common elements will definitely work. if we put a 9 in the middle circle of the venn diagram above, then we have the following:
1, 7, 5 (imagine these numbers where you see '2, 1, 3' in the link)
8, 9, 9 (imagine these numbers where you see '0, 23, 4' in the link)
9 (where you see '191' in the link)
but, if you put an 8 in the middle circle, you get the following, which also works:
0, 8, 4
9, 8, 10
8
so, insufficient.
you can also place an 'x' in the middle circle. then you'll get
(blank for now) (16 - x) (blank for now)
(17 - x) (x) (18 - x)
(blank for now)
now, fill in the rest.
(a) must add to 25 elements. therefore, upper-left blank + (16 - x) + (17 - x) + (x) = 25.
upper-left blank + 33 - x = 25
upper-left blank = x - 8
similarly
(b) must add to 30 elements. therefore, upper-right blank + (16 - x) + (18 - x) + (x) = 30
upper-right blank + 34 - x = 30
upper-right blank = x - 4
similarly
(c) must add to 35 elements. therefore, bottom blank + (17 - x) + (18 - x) + (x) = 35
bottom blank + 35 - x = 35
bottom blank = x
therefore, the finished diagram is
(x - 8) (16 - x) (x - 4)
(17 - x) (x) (18 - x)
(x)
ANY x that makes all these numbers nonnegative is acceptable. this means that x can actually be any number between 8 and 16, inclusive. (if x < 8, then the 'x - 8' is negative; if x > 16, then '16 - x' is negative.)
insufficient.
answer (a).
