# Section-1: Venn Diagrams Solved QuestionLogical Reasoning Discussion

Common Information

At the annual sports meet, 106 sportsmen participated in hockey, 122 in football and 120 in cricket. It is known that 48 participated in hockey and football and 70 in football and cricket. A total of 200 sportsmen participated in the meet.

 Q. Common Information Question: 2/2 How many sportsmen did not participate in either cricket or football? Assume data from the common data question 1 to be valid.
 ✖ A. 25 ✖ B. 27 ✔ C. 28 ✖ D. 30

Solution:
Option(C) is correct

To solve this question set, let us introduce a variable x that represents the set of players that participated in all three sports. In the Venn diagram, this set will be represented by the area that is common to all three circles. In symbolic form, it means:

n(H FC) = x

Now, it is given that the number of students that participated in both Hockey and Football is 48.

This means that the number of students that participated only in both Football and Hockey and not Cricket = 48 – x

Similarly, the number of students that participated in Football and Cricket but not in Hockey = 70 – x

In this way, we can label all the sections of the Venn diagram in terms of known numbers and the variable x.

After incorporating data from the first question (that is said to be valid for the second question as well), this is the completed Venn diagram:

Clearly from the diagram the number of sportsmen who did not participate in either cricket or football = (x + 4) + 10 = 28

## (2) Comment(s)

Saikat Basu
()

Hi,

I am not able to understand this solution. As per my calculations, 106 (for H) - [ 14 (for all three) + 56 (for C and H, excluding F) + 34 (for H and F, excluding C) + 40 (for C and F, excluding H) ] + 10 (for Non players) which is summing up to 12.

Kindly suggest where I am going wrong.

Regards,

Saikat Basu

Ani
()

Hi saikat,

your approach for this question is a little bit wrong buddy, why are you subtracting from total number of Hockey players.

well lets see if you get any of my approaches that i have tried, and the easiest one is given already, still.

here is another one, correct me if i m wrong.

As the question goes, we need to find how many players didn't play either cricket or football, so find out how many players only played hockey.

so,

(200)total number of players - players [ only football (18)+only cricket (10) +(14) all the three+cricket and football (56)+cricket and hockey (40)+football and hockey (34)]

200 - 172 = 28 players.

but in this total 28 players who play only hockey is 18 because 10 players don't play anything. still ans for the question will be 28.