Question 12: In a secondary school with 125 students participate in at least one of the following sports: cricket, football, or hockey.
* 60 students play cricket.
* 70 students play football.
* 40 students play hockey.
* 25 students play both cricket and football.
* 15 students play both football and hockey.
* 10 students play both cricket and hockey.
\((a)\) How many students play all three sports?
Solution:
Let, \(C=\text{Participants in cricket}\; ;\)\(F=\text{Participants in football}\; ;\)\(\;H=\text{Participants in hockey}\)
\(n(C \cup F \cup H)=125\)
\(n(C)=60\)
\(n(F)=70\)
\(n(H)=40\)
\(n(C\cap F)=25\)
\(n(F\cap H)=15\)
\(n(C \cap H)=10\)
\(n(C\cap F \cap H)=?\)
We know that
\(n(C\cup F\cup H)=\ \)\(n(C)+n(F)+n(H)\ \)\(−\ n(C\cap F)−n(F\cap H)\ \)\(−\ \)\(n(C\cap H)+n(C\cap F\cap H)\)
\(125=\ \)\(60+70+40\ \)\(−\ 25−15\ \)\(−\ \)\(10+n(C\cap F\cap H))\)
\(125=\ \)\(120+n(C\cap F\cap H)\)
\(n(C\cap F\cap H)=125-120\)
\(n(C\cap F\cap H)=5\)
\(\boxed{5 \text{ students play all three sports}}\)