Three of the events at a track meet were the 100 m dash, the 200 m das

Question

Three of the events at a track meet were the 100 m dash, the 200 m dash, and the 100 m hurdles. Sixteen runners qualified for and competed in each event. Six competed only in the 100 m dash, 7 only in the 200 m dash, and 9 only in the 100 m hurdles. Additionally, 5 competed in both dashes, 3 competed in both 100 m events, and 2 competed in the 200 m dash and 100 m hurdles. How many competitors participated in all three events?

(A) 0
(B) 2
(C) 3
(D) 6
(E) 12

(A) 0
(B) 2
(C) 3
(D) 6
(E) 12

Nidzo · Answer

Formula for 3 overlapping sets:  Total = (Sum of individual groups) - (Sum of two-group groups) - 2*(Sum of three-group overlaps) + Neither

As all 16 ran in the events, we do not have a "neither" group. Let the sum of the three-group overlaps = x

16 = (6 + 7 + 9) - (5 + 3 + 2) - 2*(x)

16 = 22 - 10 - 2x

16 = 12 - 2x

4 = 2x

2 = x

Answer B

archana21 · Answer

From the sentence in bold, I understand that 16 runners played in each event. 16 is not the total number of players. Please correct if I misinterpreted it.

using this, if the number of players playing all three events is c
100m dash 16 = 6 + 5-c + 3-c + c
c = -2

but, why is this incorrect?!

archana21 · Answer

Bunuel  - could you please clarify if the total number is 16 or number in each event is 16.

ShaikhMoice · Answer

Experts kindly provide the solution.
Thanks in advance

abhishekmayank · Answer

Should not the solution be:

Total = (Only in one event) + (In two events) - 2*(All the events)
16 = (6+7+9) + (5+3+2) - 2*(all the events)

Where am I getting wrong ?

GMATist1 · Answer

Total = Only in one event + 2*(in two events) + 3*(all the events)
48 = (6+7+9) + 2*(5+3+2) + 3*(all three events)

ShreyasJavahar · Answer

Firstly, it is only 16 athletes in total competing, not 16 in each event. This is evidenced by the fact that we don't actually have different sets of 16 athletes competing in each of the events; we have 6 in the 100m dash, 7 in the 200m dash, and 9 in the 100m hurdles. That being said, the phrase "Sixteen runners qualified for and competed in each event." could have been worded better.
Moving on, 9+7+6 = 22
22 - 16 = 6. So we have 6 athletes competing in at least 2 events. Let's shift our focus to these 6 athletes, let's call them A, B, C, D, E, and F.

1) 5 competed in both dashes -> Let's say A, B, C, D, and E competed in the dashes.
2) 3 competed in both 100m events -> Let's say D, E, and F competed in the 100m events. 
3) 2 competed in the 200 m dash and 100 m hurdles -> Notice that we already have D and E competing in both the 200m dash and the 100m hurdles, and they also compete in the 100m dash, so it must be these 2 competing in all three events.

Note : We cannot take more than 2 athletes from step (1) in step (2) as that would imply that 3 athletes competed in both the 200m dash and the 100m hurdles. Having three athletes in step (3) would be in violation of the conditions provided, therefore, F must be one of the athletes in step (2).

ShreyasJavahar · Answer

The 6, 7 and 9 are not values for the "only in one event" category. The wording of the problem does lead one to believe that the values are for only one event, but it does not logically follow for the following reasons :
1)Sixteen runners qualified for and competed in each event. We have 16 runners in total here, not 3 different sets of 16 runners. The "competed in each event" mostly means that the number of runners who did not compete in any event is zero.
2)Six competed only in the 100 m dash, 7 only in the 200 m dash, and 9 only in the 100 m hurdles. If it were true that 16 runners qualified and competed in "each event", that would mean we should have had 16 runners competing in every event, that is however not the case. The "only" here again is questionable, but I suppose we can take it to mean there were only 6 competitors in the 100 m dash, only 7 in the 200 m dash and only 9 in the hurdles.

The wording isn't particularly great, but what it does ultimately mean is that we have 16 runners competing in 3 events, under the conditions laid out in the prompt. 
You can refer to this link for more information about the formulae :  https://gmatclub.com/forum/formulae-for-3-overlapping-sets-69014.html#p729340

ShreyasJavahar · Answer

Hi, this isn't the right formula. You did get the answer in this particular case, but you might not be as lucky using the same formula elsewhere.
You can refer to this link :  https://gmatclub.com/forum/formulae-for-3-overlapping-sets-69014.html#p729340

ShilpiAgnihotrii · Answer

Nidzo, the soln which u have posted will give answer as -2 and NOT 2. pls revisit your solution.

Krunaal · Answer

The quickest way is to draw the Venn diagram as attached in the image file below.

? = 16 - 6 - 5 - 3

You get  2

You can check it for any event, the total will come to 16.

bumpbot · Answer

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Three of the events at a track meet were the 100 m dash, the 200 m das

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