# Solution

Okay, this is how I have solved it; please let me know whether I have done it right.

Let us start by drawing the Venn diagram as above. We draw three intersecting circles, with each circle representing one of the sports - cricket, hockey and football. The region coloured scarlet red is common to all the three circles, and represents the set of students who play all the three games. We have to find out how many students play all the three games; let this number be "x". It is better to mark this region first.

The region coloured green represents the set of students who play both cricket and football but not hockey (this region is common only to cricket and football). We have been given that the number of students who play both cricket and football = 15. This does not mean that all these students hate hockey; among these 15 students will be a few who play hockey also (that is, among these 15 students will be some who play all the three games). Now there are "x" students who play all the three games. Therefore, the number of students who play both cricket and football, but not hockey, is 15-x. By the way, x has to be less than 15; suppose x > 15, this implies number of students who play all the three games is greater than the number of students who play cricket and football. This is not possible.

Likewise, the region coloured yellow is common only to the cricket and hockey circles, and hence represents the set of students playing cricket and hockey but not football. We have been given that 18 students like to play both cricket and hockey. Therefore, by the same argument we can say that 18-x students play the two games of cricket and hockey but not football.

The region coloured pink is common only to the hockey and football circles and, therefore, represents the set of students who like to play hockey and football but not cricket. Eleven students like to play both football and hockey. Using the same argument as above again, we can state that the number of students who like to play hockey and football but not cricket is 11-x.

Now, we have been told that 30 students like cricket. So, the sum of all the numbers within the cricket circle should be 30. Thus, the number of students who like cricket = number of students who like only cricket + number of students who like cricket and football but not hockey (green region) + number of students who like all the three games (scarlet red region) + number of students who like cricket and hockey but not football (yellow region).

Therefore, 30 = number of students who like only cricket + (15-x) + x + (18-x). So, the number of students who like only cricket = 30 - 15 + x - x - 18 + x. Therefore, number of students who like only cricket = (x - 3).

Similarly, we have been told that 25 students like to play football. Therefore, the number of students who like to play only football works out to be (x - 1).

Thirty-two students like to play hockey. The number of students who like to play only hockey works out to be (x + 3).

There are a total of 50 students and each student plays one game or the other. Hence the sum of the numbers within all the three circles should be 50.

Therefore, (x - 3) + (15 - x) + (x - 1) + x + (18 - x) + (11 - x) + (x + 3) = 50.

Therefore, x + 43 = 50. So, x = 7.

Thus:

The number of students who play all the three games = x = 7.

The number of students who play only football = x - 1 = 7 - 1 = 6.

The number of students who play only hockey = x + 3 = 7 + 3 = 10.

Did I get it right?

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• Central Institute of Fisheries Nautical and Engineering Training - www.cifnet.nic.in
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• Tata Institute of Social Sciences, Mumbai - www.tiss.edu